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Exotic ˉD()sD() molecular states and scˉqˉc tetraquark states with JP=0+, 1+, 2+

  • We have calculated the mass spectra for the ˉD()sD() molecular states and scˉqˉc tetraquark states with JP=0+,1+,2+. The masses of the axial-vector ˉDsD, ˉDsD molecular states and 1[sc]0[ˉqˉc], 0[sc]1[ˉqˉc] tetraquark states are predicted to be approximately 3.98 GeV, in good agreement with the mass of Zcs(3985) from BESIII. In both the molecular and diquark-antidiquark scenarios, our results suggest that there may exist two almost degenerate states, as the strange partners of X(3872) and Zc(3900). We propose to carefully examine Zcs(3985) in future experiments to verify this. One may also search for more hidden-charm four-quark states with strangeness in not only the open-charm ˉD()sD() channels but also the hidden-charm channels ηcK/K, J/ψK/K.
  • Very recently, the BESIII Collaboration announced a new structure near the DsD0 and DsD0 thresholds in the K+ recoil-mass spectra in e+eK+(DsD0+DsD0) [1]. The pole mass and width of this Zcs(3985) resonance were measured as (3982.5+1.82.6±2.1)MeV and (12.8+5.34.4±3.0)MeV, respectively. Decaying into DsD0 and DsD0 in the S-wave, the spin-parity of Zcs(3985) is assumed to favor JP=1+ and the quark content cˉcsˉu [1]. It will be the first candidate for the hidden-charm four-quark state with strangeness.

    In previous theoretical investigations of the hidden-charm four-quark states with strangeness, the compact tetraquark configuration scˉqˉc has been studied in the color-magnetic interaction method [2] and QCD sum rules [3-10]. In Ref. [11], the authors investigated charged charmonium-like structures with hidden-charm and open-strange channels using the initial single chiral particle emission mechanism. Their results suggested the existence of enhancement structures near the thresholds of ˉD()D()s. In Ref. [12], an axial-vector hidden-charm DD+sDD+s molecular state was also predicted to exist. Possible DˉDs0(2317) and DˉDs1(2460) molecules were studied in Ref. [13], in which the results disfavor the existence of such states.

    A hadronic molecule is composed of two color-singlet hadrons by exchanging light mesons. This is a very useful configuration to study the nature of some exotic XYZ states and pentaquark states [14-19]. Because Zcs(3985) lies very close to the mass thresholds of DsD0 and DsD0, it is naturally studied in a molecular picture [20-27], as a partner state of Zc(3900) discovered by BESIII [28]. It is also explained as a compound mixture of four different four-quark configurations [29], or a reflection structure of the charmed-strange meson Ds2(2573) [30]. In addition, the production mechanisms of the hidden-charm four-quark states with strangeness have been studied in Refs. [31, 32]. In Ref. [4], the authors studied the decay width of the DsˉD/DsˉD by calculating the three-point correlation functions in QCD sum rules. Their result for the total width suffers from a large uncertainty, although its central value is consistent with the experimental result of Zcs(3985). Such large uncertainty for the total width originates from the square of the form factors, which is inherent and difficult to be reduced using the method of three-point QCD sum rules. We also refer to the works [33-39] for recent studies on Zcs(3985) using other methods. In this work, we shall study the exotic ˉD()sD() molecular states and scˉqˉc tetraquark states with JP=0+,1+,2+ using the method of QCD sum rules [40-42].

    The paper is organized as follows. In Sec. II, we construct the interpolating currents for the ˉD()sD() molecular systems and scˉqˉc tetraquark systems with JP=0+,1+, and 2+. In Sec. III, we calculate the correlation functions and spectral densities for these interpolating currents. We extract the masses for the ˉD()sD() molecular states and scˉqˉc tetraquark states by performing the QCD sum rule analyses in Sec. IV. The last section presents a summary and discussion.

    The color structures of a molecular field [qˉQ][Qˉq] and a tetraquark field [qQ][ˉQˉq] can be written via the SU(3) symmetry,

    (3¯3)[qˉQ](3¯3)[Qˉq]=(18)[qˉQ](18)[Qˉq]=(11)(18)(81)(88)=188(18810¯1027),(33)[qQ](¯3¯3)[ˉQˉq]=(6¯3)[qQ](3¯6)[ˉQˉq]=(6¯6)(¯33)(63)(¯3¯6)=(1827)(18)(810)(8¯10),

    (1)

    in which the color singlet structures come from the \left({\bf  1}_{[q \bar{Q}]} \otimes{\bf 1}_{[Q \bar{q}]}\right) and (8[qˉQ]8[Qˉq]) terms for the molecular field and from the (6[qQ]¯6[ˉQˉq]) and (¯3[qQ]3[ˉQˉq]) terms for the tetraquark field. In this work, we shall consider the molecular and tetraquark interpolating currents with color structures (1[qˉQ]1[Qˉq]) and (¯3[qQ]3[ˉQˉq]), respectively. To study the lowest lying molecular and tetraquark states, we use only S-wave mesonic and diquark fields to construct the molecular and tetraquark currents with the angular momentum L=0 between two mesonic fields and also between two diquark fields. Finally, we obtain the ˉD()sD() molecular interpolating currents as

    J1=(ˉcaγ5sa)(ˉqbγ5cb),JP=0+,J2=(ˉcaγμsa)(ˉqbγμcb),JP=0+,J1μ=(ˉcaγμsa)(ˉqbγ5cb),JP=1+,J2μ=(ˉcaγ5sa)(ˉqbγμcb),JP=1+,J3μ=(ˉcaγαsa)(ˉqbσαμγ5cb),JP=1+,J4μ=(ˉcaσαμγ5sa)(ˉqbγαcb),JP=1+,Jμν=(ˉcaγμsa)(ˉqbγνcb),JP=2+,

    (2)

    and the scˉqˉc tetraquark interpolating currents as

    η1=sTaCγ5cb(ˉqaγ5CˉcTbˉqbγ5CˉcTa),JP=0+,η2=sTaCγμcb(ˉqaγμCˉcTbˉqbγμCˉcTa),JP=0+,η1μ=sTaCγμcb(ˉqaγ5CˉcTbˉqbγ5CˉcTa),JP=1+,η2μ=sTaCγ5cb(ˉqaγμCˉcTbˉqbγμCˉcTa),JP=1+,η3μ=sTaCγαcb(ˉqaσαμγ5CˉcTbˉqbσαμγ5CˉcTa),JP=1+,η4μ=sTaCσαμγ5cb(ˉqaγαCˉcTbˉqbγαCˉcTa),JP=1+,ημν=sTaCγμcb(ˉqaγνCˉcTbˉqbγνCˉcTa),JP=2+,

    (3)

    in which a, b denote color indices and q is an up or down quark. The mesonic field ˉqaσαμγ5qa in J3μ and J4μ can couple to both the vector channel JP=1(ˉqaσijγ5qa) and axial-vector channel JP=1+ (ˉqaσ0iγ5qa). We pick out its S-wave vector component by multiplicating a vector mesonic field ˉqγαq, so that the molecular operators carry positive parity. A similar situation occurs for the tetraquark currents η3μ and η4μ. The molecular currents in Eq. (2) are not independent of the diquark-antidiquark currents in Eq. (3). Actually, a molecular current can be rewritten in terms of a sum over diquark-antidiquark currents via Fierz transformation with some suppression factors. In this work, we shall establish both the mass spectra for these two different configurations. Using the interpolating currents in Eqs. (2) and (3), we shall study the masses for the ˉD()sD() molecular states and scˉqˉc tetraquark states in the following sections.

    In this section, we study the two-point correlation functions of the scalar, axial-vector, and tensor interpolating currents above. For the scalar currents, the correlation function is

    Π(p2)=id4xeipx0|T[J(x)J(0)]|0,

    (4)

    and that for the axial-vector current is

    Πμν(p2)=id4xeipx0|T[Jμ(x)Jν(0)]|0.

    (5)

    The correlation function Πμν(p2) in Eq. (5) can be rewitten as

    Πμν(p2)=(pμpνp2gμν)Π1(p2)+pμpνp2Π0(p2),

    (6)

    where Π0(p2) and Π1(p2) are the scalar and vector current polarization functions corresponding to the spin-0 and spin-1 intermediate states, respectively. The correlation function for the tensor current Jμν(x) is

    Πμν,ρσ(p2)=id4xeipx0|T[Jμν(x)Jρσ(0)]|0,

    (7)

    which can be expressed as

    Πμν,ρσ(p2)=(ημρηνσ+ημσηνρ23ημνηρσ)Π2(p2)+,

    (8)

    where

    ημν=pμpνp2gμν,

    (9)

    and Π2(p2) is the tensor current polarization functions related to the spin-2 intermediate states; represents other spin-0 or spin-1 states.

    At the hadronic level, the correlation function can be described via the dispersion relation

    Π(p2)=(p2)Nπ4m2cImΠ(s)sN(sp2iϵ)ds+N1n=0bn(p2)n,

    (10)

    where bn is the subtraction constant. In QCD sum rules, the imaginary part of the correlation function is defined as the spectral function

    ρ(s)=1πImΠ(s)=f2Hδ(sm2H)+QCD continuum and higher states,

    (11)

    in which the “pole plus continuum parametrization” is used. The parameters fH and mH are the coupling constant and mass of the lowest-lying hadronic resonance H, respectively

    0|J|H=fH,0|Jμ|H=fHϵμ,0|Jμν|H=fHϵμν

    (12)

    with the polarization vector ϵμ and polarization tensor ϵμν.

    We can calculate the correlation function Π(p2) and spectral density ρ(s) by means of operator product expansion (OPE) at the quark-gluon level. To evaluate the Wilson coefficients, we adopt the propagator of a light quark in coordinate space and the propagator of a heavy quark in momentum space

    iSabq(x)=iδab2π2x4ˆx+i32π2λnab2gsGnμν1x2(σμνˆx+ˆxσμν)δabx212ˉqgsσGqmqδab4π2x2+iδabmq(ˉqq)48ˆximqˉqgsσGq)δabx2ˆx1152,iSabQ(p)=iδabˆpmQ+i4gsλnab2Gnμνσμν(ˆp+mQ)+(ˆp+mQ)σμν12+iδab12g2sGGmQp2+mQˆp(p2m2Q)4,

    (13)

    where q is the u, d, or s quark, and Q represents the c or b quark. The superscripts a,b denote the color indices, and ˆx=xμγμ,ˆp=pμγμ. In this work, we calculate the Wilson coefficients up to dimension eight condensates at the leading order in αs. In Ref. [43], the NLO perturbative corrections to the correlation functions for the scˉqˉc tetraquark systems have been studied, and their results show that such contributions are numerically small. The spectral densities for the interpolating currents in Eqs. (2) and (3) are evaluated and listed in appendix A. The tetraquark currents η1(x), η2(x), η1μ(x), and η2μ(x) are the same as η2(x), η4(x), η2μ(x), and η4μ(x) for the scˉqˉb systems in Ref. [44], by replacing the bottom quark with the charm quark bc. Thus, we do not list the spectral densities for these four tetraquark currents in appendix A. To improve the convergence of the OPE series and suppress the contributions from the continuum and higher states region, the Borel transformation is applied to the correlation function at both the hadron and the quark-gluon levels. The QCD sum rules are then established as

    Lk(s0,M2B)=f2Hm2kHem2H/M2B=s04m2cdses/M2Bρ(s)sk,

    (14)

    in which MB represents the Borel mass introduced by the Borel transformation, and s0 is the continuum threshold. The mass of the lowest-lying hadron can be thus extracted as

    mH(s0,M2B)=L1(s0,M2B)L0(s0,M2B),

    (15)

    which is the function of the two parameters M2B and s0. We shall discuss in detail how to obtain suitable parameter working regions in QCD sum rule analyses in the next section.

    In this section, we perform the QCD sum rule analyses for the ˉD()sD() molecular and scˉqˉc tetraquark systems using the interpolating currents in Eqs. (2) and (3). We use the values of quark masses and various QCD condensates as follows [45-53]

    mu(2GeV)=(2.2+0.50.4)MeV ,md(2GeV)=(4.7+0.50.3)MeV,mq(2GeV)=(3.5+0.50.2)MeV,ms(2GeV)=(95+93)MeV,mc(mc)=(1.275+0.0250.035)GeV,mb(mb)=(4.18+0.040.03)GeV,ˉqq=(0.24±0.03)3GeV3,ˉqgsσGq=M20ˉqq,M20=(0.8±0.2)GeV2,ˉss/ˉqq=0.8±0.1,g2sGG=(0.48±0.14)GeV4,

    (16)

    where the u,d,s quark masses are the current quark masses obtained in the ¯MS scheme at the scale μ=2 GeV. We use the running mass in the ¯MS scheme for the charm quark, which is different from the value of the pole quark mass. Various reports show that the use of the ¯MS mass of the charm quark can lead to very good predictions for the masses of XYZ states in the framework of QCD sum rules [15, 54].

    To establish a stable mass sum rule, one should find appropriate parameter working regions first, i.e, the continuum threshold s0 and the Borel mass M2B. The threshold s0 can be determined via the minimized variation of the hadronic mass mH with the Borel mass M2B. The lower bound on the Borel mass M2B can be fixed by requiring a reasonable OPE convergence, while its upper bound is determined through a sufficient pole contribution. The pole contribution is defined as

    PC(s0,M2B)=L0(s0,M2B)L0(,M2B),

    (17)

    where L0 has been defined in Eq. (14).

    We use the ˉDsD molecular current J2(x) with JP=0+ as an example to show the details of the numerical analysis. For this current, the dominant non-perturbative contribution to the correlation function comes from the quark condensate ˉqq and ˉss. In Fig. 1, we show the contributions of the perturbative term and various condensate terms to the correlation function. It is clear that the Borel mass M2B should be large enough to ensure the convergence of the OPE series. Here, we require the highest dimension condensate contribution to be less than 10%,

    Figure 1

    Figure 1.  (color online) OPE convergence for the ˉDsD molecular current J2(x) with JP=0+.

    ΠˉqqˉqgsσGq(M2B,)Π(M2B,)<10%,

    (18)

    which results in M2B2.6GeV2.

    As mentioned above, the variation of the output hadron mass mH with M2B should be minimized to obtain the optimized value of the continuum threshold s0. We show the variations of mH with s0 and M2B in Fig. 2, from which the dependence of mH on M2B can be minimized at s020.5GeV2. Requiring the pole contribution to be larger than 30%, the upper bound on M2B can then be determined to be 3.4GeV2. The working region of the Borel parameter for the scalar ˉDsD molecular current J2(x) is thus 2.6M2B3.4GeV2. As shown in Fig. 2, the mass sum rules are established to be very stable in these parameter regions, and the hadron mass for the ˉDsD molecule with JP=0+ can be obtained as

    Figure 2

    Figure 2.  (color online) Variations of mH with s0 and M2B corresponding to the current J2(x) in the ˉDsD system with JP=0+.

    mˉDsD,0+=4.11±0.14GeV,

    (19)

    in which the error comes from the uncertainties of the continuum threshold s0, Borel mass MB, the various condensates, and quark masses. After performing similar analyses, we obtain the numerical results for all the other interpolating currents in Eqs. (2) and (3) and present them in Table 1.

    Table 1

    Table 1.  Numerical results for the ˉD()sD() molecular and diquark-antiquark scˉqˉc tetraquark systems.
    System Current JP s0/GeV2 M2B/GeV2 mH/GeV PC (%)
    ˉDsD J1 0+ 18.0 ± 2.0 1.6 ~ 3.6 3.74 ± 0.13 52.5
    ˉDsD J2 0+ 20.5 ± 2.0 2.6 ~ 3.4 4.11 ± 0.14 42.4
    ˉDsD J1μ 1+ 20.7 ± 2.0 2.1 ~ 2.5 3.99 ± 0.12 68.2
    ˉDsD J2μ 1+ 20.5 ± 2.0 2.1 ~ 2.5 3.97 ± 0.11 67.7
    ˉDsD J3μ 1+ 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.1
    ˉDsD J4μ 1+ 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.0
    ˉDsD Jμν 2+ 23.0 ± 2.0 2.8 ~ 4.3 4.34 ± 0.13 48.7
    0[sc]0[ˉqˉc] (spin-spin) η1 0+ 18.0 ± 2.0 2.1 ~ 3.1 3.84 ± 0.15 46.3
    1[sc]1[ˉqˉc] η2 0+ 20.0 ± 2.0 2.6 ~ 3.2 4.13 ± 0.17 35.6
    1[sc]0[ˉqˉc] η1μ 1+ 19.0 ± 2.0 2.5 ~ 3.3 3.98 ± 0.16 41.0
    0[sc]1[ˉqˉc] η2μ 1+ 19.0 ± 2.0 2.5 ~ 3.3 3.97 ± 0.15 41.6
    1[sc]1[ˉqˉc] η3μ 1+ 22.0 ± 2.0 2.9 ~ 3.6 4.28 ± 0.14 40.9
    1[sc]1[ˉqˉc] η4μ 1+ 22.0 ± 2.0 2.9 ~ 3.6 4.28 ± 0.14 41.1
    1[sc]1[ˉqˉc] ημν 2+ 23.0 ± 2.0 2.8 ~ 4.3 4.33 ± 0.13 46.4
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    In Table 1, the mass of the scalar ˉDsD molecular state is predicted to be slightly below the open-charm threshold TˉDsD=3.84 GeV, implying that it can only decay into the hidden-charm channel ηcK. The scalar ˉDsD state is predicted to be very close to TˉDsD=4.12 GeV; however, it can decay into ˉDsD and ηcK final states kinematically in the S-wave. The masses for the ˉDsD molecular states with JP=1+,2+ are significantly above the corresponding open-charm thresholds.

    The masses obtained from the axial-vector molecular currents J1μ and J2μ are mˉDsD,1+=(3.99±0.12) GeV and mˉDsD,1+=(3.97±0.11) GeV, which are almost degenerate with each other. One may wonder whether these two currents J1μ and J2μ could couple to the same physical molecular state or not. In QCD sum rules, this can be specified by studying the following off-diagonal correlation function

    ΠM12μν(p2)=id4xeipx0|T[J1μ(x)J2ν(0)]|0.

    (20)

    Our calculation shows that this off-diagonal correlation function ΠM12μν(p2)=0 at the leading order of αs for the axial-vector molecular currents J1μ and J2μ, including the perturbative term and all contributions from various non-perturbative condensates. According to Ref. [43], the NLO perturbative correction is numerically small; thus, ΠM12μν(p2) is still negligible compared with the diagonal correlators ΠM11μν(p2) and ΠM22μν(p2) at the next leading order of αs. This result implies that J1μ and J2μ may couple to different physical states.

    We also study the scˉqˉc tetraquark systems with JP=0+,1+,2+. In Fig. 3, we show the variations of the tetraquark mass with s0 and M2B for the current η1μ(x) with JP=1+; the mass sum rules are very stable and reliable in the chosen parameter regions. Regarding the interpolating currents in Eq. (3), we collect the numerical results for these scˉqˉc tetraquark systems in Table 1. It is shown that the mass spectra for the scˉqˉc tetraquarks are very similar to the ˉD()sD() molecular states. For the axial-vector scˉqˉc tetraquark systems, the extracted masses from η1μ(x) and η2μ(x) are almost the same as the ˉDsD and ˉDsD molecular states, which are consistent with the mass of Zcs(3985) from BESIII [1]. It is interesting to examine the off-diagonal correlation function for η1μ(x) and η2μ(x)

    Figure 3

    Figure 3.  (color online) Variations of mH with s0 and M2B for the current η1μ(x) in the scˉqˉc tetraquark system with JP=1+.

    ΠT12μν(p2)=id4xeipx0|T[J1μ(x)J2ν(0)]|0.

    (21)

    The calculation indicates that the perturbative term and the quark condensate terms in ΠT12μν(p2) are equal to zero, This off-diagonal correlation function ΠT12μν(p2) is very small, suggesting that the currents η1μ(x) and η2μ(x) cannot strongly couple to the same physical state.

    To study the hidden-charm four-quark systems with strangeness, we have calculated the mass spectra for the ˉD()sD() molecular states and scˉqˉc tetraquark states with JP=0+,1+,2+ in the framework of QCD sum rules. We construct the corresponding molecular and tetraquark interpolating currents and calculate their two-point correlation functions and spectral densities up to dimension eight condensates at the leading order of αs. The quark condensates are found to be the most important non-perturbative contribution to the correlation functions for both molecular and tetraquark systems.

    One may wonder if the two-meson scattering states can contribute to the correlation functions in our calculations. In general, the interpolating currents can couple to all structures with the same quantum numbers, including resonances, two-meson scattering states, and continuum. Thus, these structures will give contributions to the correlation functions. However, it has been demonstrated that the two-meson scattering states cannot saturate the QCD sum rules, while only exotic four-quark states can saturate the QCD sum rules. Moreover, the contributions from the two-meson scattering states to the correlation functions are numerically negligible [43, 55].

    Our results show that the masses of the axial-vector ˉDsD, ˉDsD molecular states and the scˉqˉc tetraquark states from η1μ, η2μ are calculated in good agreement with the mass of Zcs(3985). In the present calculations, it is difficult to distinguish the nature of Zcs(3985) from the molecular and diquark-antidiquark configurations. In both the molecular and diquark-antidiquark pictures, our results suggest that there may exist two almost degenerate states, as the strange partners of X(3872) and Zc(3900). We propose to carefully examine Zcs(3985) in future experiments to verify this. One can search for more hidden-charm four-quark states with strangeness in not only the open-charm ˉD()sD() channels but also the hidden-charm channels ηcK/K, J/ψK/K.

    Note added: Since we finished this work, the LHCb Collaboration has reported two new charged resonances, Zcs(4000)+ and Zcs(4220)+, in the J/ψK+ final states [56]. Their masses and decay widths are measured to be MZcs(4000)+=4003±6+414 MeV and ΓZcs(4000)+=131±15±26 MeV, and MZcs(4220)+=4216±24+4330 MeV and ΓZcs(4220)+=233±52+9773 MeV, respectively, while their spin-parity quantum numbers are identified to prefer JP=1+. These masses and spin-parity are consistent with the axial-vector ˉDsD (ˉDsD), ˉDsD molecular states and 1[sc]0[ˉqˉc] (0[sc]1[ˉqˉc]), 1[sc]1[ˉqˉc] (1[sc]1[ˉqˉc]) tetraquark states that we have predicted in Table 1.

    According to the observation of the LHCb, the decay width of Zcs(4000) is much larger than that of Zcs(3985) observed by BESIII [1]. LHCb found no evidence that Zcs(4000) and Zcs(3985) are the same state, although their masses are very close to each other. If this is true, they may be identified as the strange partners of X(3872) and Zc(3900) with JPC=1++ and JPC=1+, respectively. We propose to carefully examine Zcs(4000) and Zcs(3985) in future experiments to verify this.

    In this appendix, we list the spectral densities for the ˉD()sD() and scˉqˉc systems with JP=0+, 1+, and 2+. The spectral density includes the perturbative term, quark condensate, gluon condensate, quark-gluon mixed condensate, four-quark condensate, and dimension eight condensate

    ρ(s)=ρ0(s)+ρ3(s)+ρ4(s)+ρ5(s)+ρ6(s)+ρ8(s),

    (22)

    in which the superscripts stand for the dimension of various condensates.

    1. Spectral densities for J1:

    ρ0aJ1(s)=32048π6αmaxαmindαβmaxβmindβ(1αβ)2α3β3(m2c(α+β)αβs)3(m2c(α+β)3αβs),

    ρ0bJ1(s)=3mc1024π6αmaxαmindαβmaxβmindβ(1αβ)2(m2c(α+β)αβs)2(2m2c(α+β)5αβs)(msα2β3+mqα3β2),

    ρ3aJ1(s)=3ˉss128π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[2(1αβ)(m2c(α+β)2αβs)mcαβ22m2cmq+(m2c(α+β)2αβs)msαβ],

    ρ3bJ1(s)=3ˉqq128π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[2(1αβ)(m2c(α+β)2αβs)mcα2β2m2cms+(m2c(α+β)2αβs)mqαβ],

    ρ4aJ1(s)=g2sGGm2c4096π6αmaxαmindαβmaxβmindβ(1αβ)2(2m2c(α+β)3αβs)(1α3+1β3),

    ρ4bJ1(s)=3g2sGGm2c2048π6αmaxαmindαβmaxβmindβ(1αβ)(m2c(α+β)αβs)(m2c(α+β)2αβs)(1α2β+1αβ2),

    ρ5aJ1(s)=3ˉsgsσGsmc256π4αmaxαmindαβmaxβmindβ[(2m2c(α+β)3sαβ)(1β2(1αβ)β2)+2mcmqβ],

    ρ5bJ1(s)=3ˉqgsσGqmc256π4αmaxαmindαβmaxβmindβ[(2m2c(α+β)3sαβ)(1α2(1αβ)α2)+2mcmsα],

    ρ5cJ1(s)=ˉqgsσGq512π4((s2m2c)mq6m2cms)14m2cs+ˉsgsσGs512π4((s2m2c)ms6m2cmq)14m2cs,

    ρ6aJ1(s)=ˉssˉqq32π2(2m2c+mcmq+mcms)14m2cs,

    Π6bJ1(M2B)=ˉssˉqqm3c32π210dα(mq1α+msα)em2cα(1α)M2B,

    Π8J1(M2B)=m4c64π210dα(ˉssˉqgsσGq+ˉqqˉsgsσGs(1α)2M2B2ˉssˉqgsσGq(1α)m2c2ˉqqˉsgsσGsαm2c)em2cα(1α)M2B,

    where

    αmin=121214m2cs,αmax=12+1214m2cs,βmin=αm2cαsm2c,βmax=1α,

    2. Spectral densities for J2:

    ρ0aJ2(s)=3512π6αmaxαmindαβmaxβmindβ(1αβ)2α3β3(m2c(α+β)αβs)3(m2c(α+β)3αβs),

    ρ0bJ2(s)=3mc512π6αmaxαmindαβmaxβmindβ(1αβ)2(m2c(α+β)αβs)2(2m2c(α+β)5αβs)(msα2β3+mqα3β2),

    ρ3aJ2(s)=3ˉss64π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[2(1αβ)(m2c(α+β)2αβs)mcαβ24m2cmq+(m2c(α+β)+2αβs)msαβ],

    ρ3bJ2(s)=3ˉqq64π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[2(1αβ)(m2c(α+β)2αβs)mcα2β4m2cms+(m2c(α+β)+2αβs)mqαβ],

    ρ4J2(s)=g2sGGm2c1024π6αmaxαmindαβmaxβmindβ(1αβ)2(2m2c(α+β)3αβs)(1α3+1β3),

    ρ5aJ2(s)=3ˉsgsσGsmc128π4αmaxαmindαβmaxβmindβ2m2c(α+β)3sαββ+3ˉqgsσGqmc128π4αmaxαmindαβmaxβmindβ2m2c(α+β)3sαβα,

    ρ5bJ2(s)=ˉqgsσGq128π4((s2m2c)mq6m2cms)14m2cs+ˉsgsσGs128π4((s2m2c)ms6m2cmq)14m2cs,

    ρ6aJ2(s)=ˉssˉqq16π2(4m2c+mcmq+mcms)14m2cs,

    Π6bJ2(M2B)=ˉssˉqqm3c16π210dα(mq1α+msα)em2cα(1α)M2B,

    Π8J2(M2B)=m4c16π210dαˉssˉqgsσGq+ˉqqˉsgsσGs(1α)2M2Bem2cα(1α)M2B,

    3. Spectral densities for J1μ:

    ρ0aJ1μ(s)=34096π6αmaxαmindαβmaxβmindβ(1αβ)2α3β3(m2c(α+β)αβs)3(m2c(α+β)5αβs),

    ρ0bJ1μ(s)=3mc1024π6αmaxαmindαβmaxβmindβ(1αβ)2(m2c(α+β)αβs)2((2m2c(α+β)5αβs)msα3β2+(m2c(α+β)4αβs)mqα2β3),

    ρ3aJ1μ(s)=3ˉss256π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[4(1αβ)(m2c(α+β)2αβs)mcα2β4m2cmq+(m2c(α+β)3αβs)msαβ],

    ρ3bJ1μ(s)=3ˉqq256π4αmaxαmindαβmaxβmindβ(m2c(α+β)αβs)[2(1αβ)(m2c(α+β)3αβs)mcαβ24m2cms+(m2c(α+β)3αβs)mqαβ],

    ρ4aJ1μ(s)=g2sGGm2c4096π6αmaxαmindαβmaxβmindβ(1αβ)2(m2c(α+β)2αβs)(1α3+1β3),

    ρ4bJ1μ(s)=g2sGGm2c4096π6αmaxαmindαβmaxβmindβ(1αβ)(m2c(α+β)αβs)(3(m2c(α+β)3αβs)αβ2(3m2c(α+β)5αβs)α2β),

    \rho_{J_{1\mu}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\alpha}\, ,

    \rho_{J_{1\mu}}^{5b}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\beta}-\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)+\frac{2m_{c}m_{s}}{\beta}\right]\, ,

    \rho_{J_{1\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{J_{1\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{J_{1\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{J_{1\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    4. Spectral densities for J_{2\mu} :

    \rho_{J_{2\mu}}^{0a}(s) = \frac{3} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

    \rho_{J_{2\mu}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \Big(\frac{(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)m_{q}}{\alpha^{2} \beta^{3}} +\frac{(m_{c}^{2}(\alpha+\beta)-4\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}\Big)\, ,

    \rho_{J_{2\mu}}^{3a}(s) = - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha\beta^{2} } -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

    \rho_{J_{2\mu}}^{3b}(s) = - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha ^{2} \beta} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

    \rho_{J_{2\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

    \rho_{J_{2\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2} \beta} -\frac{(3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s)}{\alpha\beta^{2} }\right)\, ,

    \rho_{J_{2\mu}}^{5a}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\beta}\, ,

    \rho_{J_{2\mu}}^{5b}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\alpha}-\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)+\frac{2m_{c}m_{q}}{\alpha}\right]\, ,

    \rho_{J_{2\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{J_{2\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{s}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{J_{2\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{J_{2\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    5. Spectral densities for J_{3\mu} :

    \rho_{J_{3\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

    \rho_{J_{3\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, ,

    \rho_{J_{3\mu}}^{3a}(s) = \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

    \rho_{J_{3\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \!\!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}}\! {\rm d} \alpha \!\!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1\!-\!\alpha-\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta}\!-\!\frac{12m_{c}^{2}m_{s}\! +\! 3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

    \rho_{J_{3\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

    \rho_{J_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha\beta^{2}} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2}\beta}\right)\, ,

    \rho_{J_{3\mu}}^{5a}(s) = -\frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \alpha \, ,

    \rho_{J_{3\mu}}^{5b}(s) = \frac{ \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\alpha}+\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)-\frac{6m_{c}m_{s}}{\alpha}\right]\, ,

    \rho_{J_{3\mu}}^{5c}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{J_{3\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{J_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{J_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    6. Spectral densities for J_{4\mu} :

    \rho_{J_{4\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

    \rho_{J_{4\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{2} \beta^{3}}-\frac{ \alpha \beta sm_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

    \rho_{J_{4\mu}}^{3a}(s) = \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \alpha}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

    \rho_{J_{4\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} \!{\rm d} \alpha \!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\! \beta)\! -\! \alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\! +\! \beta)\! -\! 5\alpha \beta s)m_{c}} { \alpha \beta^{2}} \! -\! \frac{12m_{c}^{2}m_{q}\! +\! 3(m_{c}^{2}(\alpha\! +\! \beta)\! -\! 3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

    \rho_{J_{4\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

    \rho_{J_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha^{2}\beta} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha\beta^{2}}\right)\, ,

    \rho_{J_{4\mu}}^{5a}(s) = -\frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \beta \, ,

    \rho_{J_{4\mu}}^{5b}(s) = \frac{ \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\beta}+\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)-\frac{6m_{c}m_{q}}{\beta}\right]\, ,

    \rho_{J_{4\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}+\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{J_{4\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{J_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{J_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    7. Spectral densities for J_{\mu\nu} :

    \rho_{J_{\mu\nu}}^{0a}(s) = -\frac{5} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\,

    \rho_{J_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

    \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3a}(s) = &- \frac{15\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}} \right. \\ &\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, ,\end{aligned}

    \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3b}(s) = & - \frac{15\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left. -\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned}

    \rho_{J_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, ,

    \begin{aligned}[b] \rho_{J_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{1}{\alpha^{2}\beta}+\frac{1}{\alpha\beta^{2}}\right)\\ &\times\left((1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)-4(m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\right)\, , \end{aligned}

    \begin{aligned}[b] \rho_{J_{\mu\nu}}^{5a}(s) = &\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned}

    \rho_{J_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{J_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{32 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{J_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{16 \pi^{2}} \int_{0}^{1} {\rm{d}} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{J_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    8. Spectral densities for \eta_{3\mu} :

    \rho_{\eta_{3\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

    \rho_{\eta_{3\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{q}}{\alpha^{2} \beta^{3}}\right)\, ,

    \rho_{\eta_{3\mu}}^{3a}(s) = \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

    \rho_{\eta_{3\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1-\alpha\!-\!\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{q}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

    \rho_{\eta_{3\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

    \rho_{\eta_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, ,

    \rho_{\eta_{3\mu}}^{5a}(s) = \frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, ,

    \rho_{\eta_{3\mu}}^{5b}(s) = -\frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, ,

    \rho_{\eta_{3\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{\eta_{3\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{\eta_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{\eta_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+2\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    9. Spectral densities for \eta_{4\mu} :

    \rho_{J_{4\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

    \rho_{\eta_{4\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, ,

    \rho_{\eta_{4\mu}}^{3a}(s) = \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

    \rho_{\eta_{4\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\! + \! \beta)\! - \!\alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{s}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

    \rho_{\eta_{4\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

    \rho_{\eta_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, ,

    \ \rho_{\eta_{4\mu}}^{5a}(s) = \frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, ,

    \rho_{\eta_{4\mu}}^{5b}(s) = -\frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, ,

    \rho_{\eta_{4\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{\eta_{4\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{\eta_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{\eta_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    10. Spectral densities for \eta_{\mu\nu} :

    \rho_{\eta_{\mu\nu}}^{0a}(s) = -\frac{5} {768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\,

    \rho_{\eta_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {384 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

    \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3a}(s) = & - \frac{5\langle\bar{s}s\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha^{2} \beta}\right.\\ &\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, , \end{aligned}

    \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3b}(s) = & - \frac{5\langle\bar{q}q\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left.-\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned}

    \rho_{\eta_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, ,

    \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{12288 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left[(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)\left(1+\frac{2(1-\alpha-\beta)^{2}}{\alpha\beta}\right) \right.\\ &\left.+\frac{4(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)(1-\alpha-\beta)(\alpha+\beta)}{\alpha\beta^{2}}\right]\, , \end{aligned}

    \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{5a}(s) =&\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned}

    \rho_{\eta_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \rho_{\eta_{\mu\nu}}^{5c}(s) = \frac{5 \left(\langle\bar{s} g_{s}\sigma\cdot G s\rangle+\langle\bar{q} g_{s}\sigma\cdot G q\rangle\right) m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{7(m_{c}^{2}(\alpha+\beta)-6 \alpha \beta s)(\alpha+5(1-\alpha+\beta))}{\alpha\beta}\, ,

    \rho_{\eta_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{24 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

    \Pi_{\eta_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{12 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

    \Pi_{\eta_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \left[\frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}}-\frac{\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{12\alpha}\right] {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

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Cited by

1. Liu, X., Chen, D., Huang, H. et al. Predictions of the strange partner of Tcc in the quark delocalization color screening model[J]. Physical Review D, 2024, 109(5): 054021. doi: 10.1103/PhysRevD.109.054021
2. Chen, K.. SU(3) breaking effect in the Zc and Zcs states[J]. Physical Review D, 2024, 109(3): 034010. doi: 10.1103/PhysRevD.109.034010
3. Agaev, S.S., Azizi, K., Barsbay, B. et al. Fully charmed resonance X(6900) and its beauty counterpart[J]. Nuclear Physics A, 2024. doi: 10.1016/j.nuclphysa.2023.122768
4. Ablikim, M., Achasov, M.N., Adlarson, P. et al. Observation of a Vector Charmoniumlike State at 4.7 GeV/c2 and Search for Zcs in e+e- →k+K-J/ψ[J]. Physical Review Letters, 2023, 131(21): 211902. doi: 10.1103/PhysRevLett.131.211902
5. Güngör, E., Sundu, H., Y. Süngü, J. et al. Possible Molecular Explanation for the Resonance Y (4500)[J]. Few-Body Systems, 2023, 64(3): 53. doi: 10.1007/s00601-023-01807-y
6. Chen, S., Li, Y., Qian, W. et al. Heavy flavour physics and CP violation at LHCb: A ten-year review[J]. Frontiers of Physics, 2023, 18(4): 44601. doi: 10.1007/s11467-022-1247-1
7. Meng, L., Wang, B., Wang, G.-J. et al. Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules[J]. Physics Reports, 2023. doi: 10.1016/j.physrep.2023.04.003
8. Wu, Y., Jin, X., Liu, R. et al. Study of double-charm and double-strange tetraquark Xcc s ¯ s ¯[J]. Physical Review D, 2023, 107(9): 094011. doi: 10.1103/PhysRevD.107.094011
9. Agaev, S.S., Azizi, K., Sundu, H. Modeling the resonance Tcs0a (2900)++ as a hadronic molecule D∗+K∗+[J]. Physical Review D, 2023, 107(9): 094019. doi: 10.1103/PhysRevD.107.094019
10. Agaev, S.S., Azizi, K., Sundu, H. On the structures of new scalar resonances Tacs0 ( 2900 )+ + and Tacs0 ( 2900 )0[J]. Journal of Physics G: Nuclear and Particle Physics, 2023, 50(5): 055002. doi: 10.1088/1361-6471/acc41a
11. Wu, Q., Chen, Y.-K., Li, G. et al. Hunting for the hidden-charm molecular states with strange quarks in B and Bs decays[J]. Physical Review D, 2023, 107(5): 054044. doi: 10.1103/PhysRevD.107.054044
12. Ablikim, M., Achasov, M.N., Adlarson, P. et al. Search for hidden-charm tetraquark with strangeness in e+e-→K+ D*+s D*0 + c.c.*[J]. Chinese Physics C, 2023, 47(3): 033001. doi: 10.1088/1674-1137/acac69
13. Chen, H.-X., Chen, W., Liu, X. et al. An updated review of the new hadron states[J]. Reports on Progress in Physics, 2023, 86(2): 026201. doi: 10.1088/1361-6633/aca3b6
14. Luo, X., Nakamura, S.X. X and Zcs in B+ →j/ψφK+ as s -wave threshold cusps and alternative spin-parity assignments to X (4274) and X (4500)[J]. Physical Review D, 2023, 107(1): L011504. doi: 10.1103/PhysRevD.107.L011504
15. Xin, Q., Wang, Z.-G., Yang, X.-S. Analysis of the X(3960) and related tetraquark molecular states via the QCD sum rules[J]. AAPPS Bulletin, 2022, 32(1): 37. doi: 10.1007/s43673-022-00070-3
16. Wang, Z.-G.. Strange cousin of Z c(4020/4025) as a tetraquark state[J]. Chinese Physics C, 2022, 46(12): 123106. doi: 10.1088/1674-1137/ac8c21
17. Meng, L., Wang, B., Wang, G.-J. et al. Are the Zcs(3985) and Zcs(4000) the same state?[J]. Nuclear and Particle Physics Proceedings, 2022. doi: 10.1016/j.nuclphysbps.2022.09.018
18. Wang, Z.-G.. Decay widths of Z cs(3985/4000) based on rigorous quark-hadron duality*[J]. Chinese Physics C, 2022, 46(10): 103106. doi: 10.1088/1674-1137/ac7cd7
19. Zhai, Q.-Y., Liu, M.-Z., Lu, J.-X. et al. Zcs (3985) in next-To-leading-order chiral effective field theory: The first truncation uncertainty analysis[J]. Physical Review D, 2022, 106(3): 034026. doi: 10.1103/PhysRevD.106.034026
20. Süngü, J.Y., Türkan, A., Sundu, H. et al. Impact of a thermal medium on newly observed Zcs(3985) resonance and its b-partner[J]. European Physical Journal C, 2022, 82(5): 453. doi: 10.1140/epjc/s10052-022-10305-0
21. Chen, H.-X.. Hadronic molecules in B decays[J]. Physical Review D, 2022, 105(9): 094003. doi: 10.1103/PhysRevD.105.094003
22. Du, M.-L., Albaladejo, M., Guo, F.-K. et al. Combined analysis of the Zc (3900) and the Zcs (3985) exotic states[J]. Physical Review D, 2022, 105(7): 074018. doi: 10.1103/PhysRevD.105.074018
23. Han, S., Xiao, L.-Y. Aspects of Zcs (3985) and Zcs (4000)[J]. Physical Review D, 2022, 105(5): 054008. doi: 10.1103/PhysRevD.105.054008
24. Chen, H., Huang, Q., Ping, R.-G. Spin and polarization analysis of the Formula Presented state[J]. Physical Review D, 2022, 105(3): 036002. doi: 10.1103/PhysRevD.105.036002
25. Ikeno, N., Molina, R., Oset, E. Zcs states from the Ds∗ D ¯ ∗ and J /ψK∗ coupled channels: Signal in B+ →j /ψφK+ decay ZCS STATES from the DS∗ D ¯ ∗ ... IKENO NATSUMI, MOLINA RAQUEL, and OSET EULOGIO[J]. Physical Review D, 2022, 105(1): 014012. doi: 10.1103/PhysRevD.105.014012
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Qi-Nan Wang, Wei Chen and Hua-Xing Chen. Exotic {\bar{ D}_{ s}^{({\bf *})}{ D}^{({\bf *})}} molecular states and { {{sc}}\bar { q}\bar { c} } tetraquark states with JP=0+, 1+, 2+[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac0b3b
Qi-Nan Wang, Wei Chen and Hua-Xing Chen. Exotic {\bar{ D}_{ s}^{({\bf *})}{ D}^{({\bf *})}} molecular states and { {{sc}}\bar { q}\bar { c} } tetraquark states with JP=0+, 1+, 2+[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac0b3b shu
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Exotic {\bar{\boldsymbol D}_{\boldsymbol s}^{({\bf *})}{\boldsymbol D}^{({\bf *})}} molecular states and { {\boldsymbol{sc}}\bar {\boldsymbol q}\bar {\boldsymbol c} } tetraquark states with JP=0+, 1+, 2+

Abstract: We have calculated the mass spectra for the \bar{D}_s^{(*)}D^{(*)} molecular states and sc\bar q\bar c tetraquark states with J^P=0^+, 1^+, 2^+. The masses of the axial-vector \bar{D}_sD^{*}, \bar{D}_s^{*}D molecular states and {\bf 1}_{\boldsymbol{[sc]}}\boldsymbol \oplus {\bf 0}_{\boldsymbol{[\bar q \bar{c}]}}, {\bf 0}_{\boldsymbol{[sc]}} \oplus {\bf 1}_{\boldsymbol{[\bar q \bar{c}]}} tetraquark states are predicted to be approximately 3.98 GeV, in good agreement with the mass of Z_{cs}(3985)^- from BESIII. In both the molecular and diquark-antidiquark scenarios, our results suggest that there may exist two almost degenerate states, as the strange partners of X(3872) and Z_c(3900). We propose to carefully examine Z_{cs}(3985) in future experiments to verify this. One may also search for more hidden-charm four-quark states with strangeness in not only the open-charm \bar{D}_s^{(*)}D^{(*)} channels but also the hidden-charm channels \eta_c K/K^\ast, J/\psi K/K^\ast.

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    I.   INTRODUCTION
    • Very recently, the BESIII Collaboration announced a new structure near the D_{s}^{-} D^{* 0} and D_{s}^{*-} D^{0} thresholds in the K^+ recoil-mass spectra in e^{+} e^{-} \rightarrow K^{+}\left(D_{s}^{-} D^{* 0}+D_{s}^{*-} D^{0}\right) [1]. The pole mass and width of this Z_{cs}(3985)^- resonance were measured as \left(3982.5_{-2.6}^{+1.8} \pm 2.1\right) {\rm{MeV}} and \left(12.8_{-4.4}^{+5.3} \pm 3.0\right) {\rm{MeV}} , respectively. Decaying into D_{s}^{-} D^{* 0} and D_{s}^{*-} D^{0} in the S-wave, the spin-parity of Z_{cs}(3985)^- is assumed to favor J^P = 1^+ and the quark content c\bar cs\bar u [1]. It will be the first candidate for the hidden-charm four-quark state with strangeness.

      In previous theoretical investigations of the hidden-charm four-quark states with strangeness, the compact tetraquark configuration sc\bar{q}\bar{c} has been studied in the color-magnetic interaction method [2] and QCD sum rules [3-10]. In Ref. [11], the authors investigated charged charmonium-like structures with hidden-charm and open-strange channels using the initial single chiral particle emission mechanism. Their results suggested the existence of enhancement structures near the thresholds of \bar{D}^{(*)}D_{s}^{(*)} . In Ref. [12], an axial-vector hidden-charm D^{*-} D_{s}^{+}-D^{-} D_{s}^{*+} molecular state was also predicted to exist. Possible D\bar{D}_{s0}^{*}(2317) and D^{*}\bar{D}_{s1}^{*}(2460) molecules were studied in Ref. [13], in which the results disfavor the existence of such states.

      A hadronic molecule is composed of two color-singlet hadrons by exchanging light mesons. This is a very useful configuration to study the nature of some exotic XYZ states and pentaquark states [14-19]. Because Z_{cs}(3985)^- lies very close to the mass thresholds of D_{s}^{-} D^{* 0} and D_{s}^{*-} D^{0} , it is naturally studied in a molecular picture [20-27], as a partner state of Z_c(3900) discovered by BESIII [28]. It is also explained as a compound mixture of four different four-quark configurations [29], or a reflection structure of the charmed-strange meson D_{s2}^\ast(2573) [30]. In addition, the production mechanisms of the hidden-charm four-quark states with strangeness have been studied in Refs. [31, 32]. In Ref. [4], the authors studied the decay width of the D_s\bar D^\ast/D_s^\ast\bar D by calculating the three-point correlation functions in QCD sum rules. Their result for the total width suffers from a large uncertainty, although its central value is consistent with the experimental result of Z_{cs}(3985)^- . Such large uncertainty for the total width originates from the square of the form factors, which is inherent and difficult to be reduced using the method of three-point QCD sum rules. We also refer to the works [33-39] for recent studies on Z_{cs}(3985) using other methods. In this work, we shall study the exotic \bar{D}_s^{(*)}D^{(*)} molecular states and sc\bar q\bar c tetraquark states with J^P = 0^+, 1^+, 2^+ using the method of QCD sum rules [40-42].

      The paper is organized as follows. In Sec. II, we construct the interpolating currents for the \bar{D}_s^{(*)}D^{(*)} molecular systems and sc\bar q\bar c tetraquark systems with J^{P} = 0^{+},1^{+} , and 2^{+} . In Sec. III, we calculate the correlation functions and spectral densities for these interpolating currents. We extract the masses for the \bar{D}_s^{(*)}D^{(*)} molecular states and sc\bar q\bar c tetraquark states by performing the QCD sum rule analyses in Sec. IV. The last section presents a summary and discussion.

    II.   INTERPOLATING CURRENTS
    • The color structures of a molecular field [q \bar{Q}][Q \bar{q}] and a tetraquark field [q Q][\bar Q \bar{q}] can be written via the SU(3) symmetry,

      \begin{aligned}[b] (\mathbf{3} \!\otimes\! \overline{\mathbf{3}})_{[q \bar{Q}]} \!\otimes\!(\mathbf{3} \!\otimes\! \overline{\mathbf{3}})_{[Q \bar{q}]} & = \!(\mathbf{1} \!\oplus\! \mathbf{8})_{[q \bar{Q}]} \!\otimes\!(\mathbf{1} \!\oplus\! \mathbf{8})_{[Q \bar{q}]} \\ &=\! (\mathbf{1} \!\otimes\! \mathbf{1}) \!\oplus\!(\mathbf{1} \!\otimes\! \mathbf{8}) \!\oplus\!(\mathbf{8} \!\otimes\! \mathbf{1}) \!\oplus\!(\mathbf{8} \!\otimes\! \mathbf{8}) \\ &=\! \mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{8} \!\oplus\!(\mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{1 0} \!\oplus\! \overline{\mathbf{1 0}} \!\oplus\! \mathbf{27})\, ,\\ (\mathbf{3} \!\otimes\! {\mathbf{3}})_{[q Q]} \!\otimes\!(\overline{\mathbf{3}} \!\otimes\! \overline{\mathbf{3}})_{[\bar Q \bar{q}]} &=\! (\mathbf{6} \!\oplus\! \overline{\mathbf{3}})_{[q Q]} \!\otimes\!(\mathbf{3} \!\oplus\! \overline{\mathbf{6}})_{[\bar Q \bar{q}]} \\ &=\! (\mathbf{6} \!\otimes\! \overline{\mathbf{6}}) \!\oplus\!(\overline{\mathbf{3}} \!\otimes\! \mathbf{3}) \!\oplus\! (\mathbf{6} \!\otimes\! \mathbf{3}) \!\oplus\!(\overline{\mathbf{3}} \!\otimes\! \overline{\mathbf{6}}) \\ &=\! (\mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{27}) \!\oplus\!(\mathbf{1} \!\oplus \!\mathbf{8}) \!\oplus\! (\mathbf{8}\! \oplus\! \mathbf{1 0}) \!\oplus\! (\mathbf{8} \!\oplus\! \overline{\mathbf{1 0}})\, , \end{aligned}

      (1)

      in which the color singlet structures come from the \left({\bf 1}_{[q \bar{Q}]} \otimes {\bf 1}_{[Q \bar{q}]}\right) and \left(\mathbf{8}_{[q \bar{Q}]} \otimes \mathbf{8}_{[Q \bar{q}]}\right) terms for the molecular field and from the \left(\mathbf{6}_{[q Q]} \otimes \overline{\mathbf{6}}_{[\bar Q \bar{q}]}\right) and \left(\overline{\mathbf{3}}_{[q Q]} \otimes \mathbf{3}_{[\bar Q \bar{q}]}\right) terms for the tetraquark field. In this work, we shall consider the molecular and tetraquark interpolating currents with color structures \left(\mathbf{1}_{[q \bar{Q}]} \otimes \mathbf{1}_{[Q \bar{q}]}\right) and \left(\overline{\mathbf{3}}_{[q Q]} \otimes \mathbf{3}_{[\bar Q \bar{q}]}\right) , respectively. To study the lowest lying molecular and tetraquark states, we use only S-wave mesonic and diquark fields to construct the molecular and tetraquark currents with the angular momentum L = 0 between two mesonic fields and also between two diquark fields. Finally, we obtain the \bar{D}_s^{(*)}D^{(*)} molecular interpolating currents as

      \begin{aligned}[b] &J_{1} = (\bar{c}_{a} \gamma_{5} s_{a})(\bar{q}_{b} \gamma_{5} c_{b} )\, , \qquad\quad J^P = 0^+\, , \\ &J_{2} = (\bar{c}_{a} \gamma_{\mu} s_{a})(\bar{q}_{b} \gamma^{\mu} c_{b} )\, , \qquad\quad J^P = 0^+\, ,\\ &J_{1\mu} = (\bar{c}_{a} \gamma_{\mu} s_{a})(\bar{q}_{b}\gamma_{5} c_{b} )\, ,\qquad\;\; J^P = 1^+\, ,\\ &J_{2\mu} = (\bar{c}_{a} \gamma_{5} s_{a})(\bar{q}_{b}\gamma_{\mu} c_{b} )\, ,\qquad\;\; J^P = 1^+ \, ,\\ &J_{3\mu} = (\bar{c}_{a} \gamma^{\alpha} s_{a})( \bar{q}_{b}\sigma_{\alpha\mu} \gamma_{5}c_{b} )\, ,\quad J^P = 1^+\, ,\\ &J_{4\mu} = (\bar{c}_{a} \sigma_{\alpha\mu}\gamma_{5} s_{a})( \bar{q}_{b} \gamma^{\alpha} c_{b} )\, , \quad J^P = 1^+\, ,\\ &J_{\mu\nu} = (\bar{c}_{a} \gamma_{\mu} s_{a})( \bar{q}_{b}\gamma_{\nu}c_{b} )\, ,\qquad\;\; J^P = 2^+\, , \end{aligned}

      (2)

      and the sc\bar q\bar c tetraquark interpolating currents as

      \begin{aligned}[b] &\eta_{1} = s_{a}^{T} C \gamma_{5} c_{b}\left(\bar{q}_{a} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma_{5} C \bar{c}_{a}^{T}\right)\, , \qquad\qquad J^P = 0^+\, , \\ &\eta_{2} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma^{\mu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\mu} C \bar{c}_{a}^{T}\right)\, , \qquad\quad\;\;\;\; J^P = 0^+\, ,\\ &\eta_{1\mu} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma_{5} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 1^+\, ,\\ &\eta_{2\mu} = s_{a}^{T} C \gamma_{5} c_{b}\left(\bar{q}_{a} \gamma^{\mu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\mu} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 1^+ \, ,\\ &\eta_{3\mu} = s_{a}^{T} C \gamma^{\alpha} c_{b}\left(\bar{q}_{a} \sigma_{\alpha\mu} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \sigma_{\alpha\mu} \gamma_{5} C \bar{c}_{a}^{T}\right)\, ,\,\, \; J^P = 1^+\, ,\\ &\eta_{4\mu} = s_{a}^{T} C \sigma_{\alpha\mu}\gamma_{5} c_{b}\left(\bar{q}_{a} \gamma^{\alpha} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\alpha} C \bar{c}_{a}^{T}\right)\, ,\quad\;\;\;\, J^P = 1^+ \, ,\\ &\eta_{\mu\nu} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma^{\nu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\nu} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 2^+\, , \end{aligned}

      (3)

      in which a , b denote color indices and q is an up or down quark. The mesonic field \bar{q}_{a}\sigma_{\alpha\mu}\gamma_{5}q_{a} in J_{3\mu} and J_{4\mu} can couple to both the vector channel J^{P} = 1^{-} ( \bar{q}_{a}\sigma_{i j}\gamma_{5}q_{a} ) and axial-vector channel J^{P} = 1^{+} ( \bar{q}_{a}\sigma_{0 i}\gamma_{5}q_{a} ). We pick out its S-wave vector component by multiplicating a vector mesonic field \bar{q}\gamma_{\alpha}q , so that the molecular operators carry positive parity. A similar situation occurs for the tetraquark currents \eta_{3\mu} and \eta_{4\mu} . The molecular currents in Eq. (2) are not independent of the diquark-antidiquark currents in Eq. (3). Actually, a molecular current can be rewritten in terms of a sum over diquark-antidiquark currents via Fierz transformation with some suppression factors. In this work, we shall establish both the mass spectra for these two different configurations. Using the interpolating currents in Eqs. (2) and (3), we shall study the masses for the \bar{D}_s^{(*)}D^{(*)} molecular states and sc\bar q\bar c tetraquark states in the following sections.

    III.   QCD SUM RULES
    • In this section, we study the two-point correlation functions of the scalar, axial-vector, and tensor interpolating currents above. For the scalar currents, the correlation function is

      \begin{array}{l} \Pi\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J(x) J^{\dagger}(0)\right]\right| 0\right\rangle\, , \end{array}

      (4)

      and that for the axial-vector current is

      \begin{array}{l} \Pi_{\mu \nu}\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{\mu}(x) J_{\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array}

      (5)

      The correlation function \Pi_{\mu\nu} (p^{2}) in Eq. (5) can be rewitten as

      \Pi_{\mu \nu}\left(p^{2}\right) = \left(\frac{p_{\mu} p_{\nu}}{p^{2}}-g_{\mu \nu}\right) \Pi_{1}\left(p^{2}\right)+\frac{p_{\mu} p_{\nu}}{p^{2}}\Pi_{0}\left(p^{2}\right)\, ,

      (6)

      where \Pi_{0}\left(p^{2}\right) and \Pi_{1}\left(p^{2}\right) are the scalar and vector current polarization functions corresponding to the spin-0 and spin-1 intermediate states, respectively. The correlation function for the tensor current J_{\mu\nu}(x) is

      \begin{array}{l} \Pi_{\mu \nu,\;\rho \sigma}\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{\mu\nu}(x) J_{\rho\sigma}^{\dagger}(0)\right]\right| 0\right\rangle\, , \end{array}

      (7)

      which can be expressed as

      \Pi_{\mu \nu,\;\rho\sigma} \left(p^{2}\right) = \left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\right) \Pi_{2}\left(p^{2}\right)+\cdots \, ,

      (8)

      where

      \eta_{\mu\nu} = \frac{p_{\mu} p_{\nu}}{p^{2}}-g_{\mu \nu},

      (9)

      and \Pi_{2}\left(p^{2}\right) is the tensor current polarization functions related to the spin-2 intermediate states; {\text{“}}\cdots{\text{”}} represents other spin-0 or spin-1 states.

      At the hadronic level, the correlation function can be described via the dispersion relation

      \Pi\left(p^{2}\right) = \frac{\left(p^{2}\right)^{N}}{\pi} \int_{4m_{c}^{2}}^{\infty} \frac{{\rm{Im}} \Pi(s)}{s^{N}\left(s-p^{2}-{\rm i} \epsilon\right)} {\rm d} s+\sum\limits_{n = 0}^{N-1} b_{n}\left(p^{2}\right)^{n}\, ,

      (10)

      where b_n is the subtraction constant. In QCD sum rules, the imaginary part of the correlation function is defined as the spectral function

      \begin{aligned}[b] \rho (s) =& \frac{1}{\pi} \text{Im}\Pi(s) = f_{H}^{2}\delta(s-m_{H}^{2})\\ &+\text{QCD continuum and higher states}\, , \end{aligned}

      (11)

      in which the “pole plus continuum parametrization” is used. The parameters f_{H} and m_{H} are the coupling constant and mass of the lowest-lying hadronic resonance H , respectively

      \begin{aligned}[b] &\left\langle 0|J| H\right\rangle = f_{H}\, , \\& \left\langle 0\left|J_{\mu}\right| H\right\rangle = f_{H} \epsilon_{\mu}\, , \\& \left\langle 0\left|J_{\mu\nu}\right| H\right\rangle = f_{H} \epsilon_{\mu\nu} \end{aligned}

      (12)

      with the polarization vector \epsilon_{\mu} and polarization tensor \epsilon_{\mu\nu} .

      We can calculate the correlation function \Pi(p^{2}) and spectral density \rho(s) by means of operator product expansion (OPE) at the quark-gluon level. To evaluate the Wilson coefficients, we adopt the propagator of a light quark in coordinate space and the propagator of a heavy quark in momentum space

      \begin{aligned}[b] {\rm i} S_{q}^{a b}(x) =& \frac{{\rm i} \delta^{a b}}{2 \pi^{2} x^{4}} \hat{x} +\frac{{\rm i}}{32 \pi^{2}} \frac{\lambda_{a b}^{n}}{2} g_{s} G_{\mu \nu}^{n} \frac{1}{x^{2}}\left(\sigma^{\mu \nu} \hat{x}+\hat{x} \sigma^{\mu \nu}\right) \\ &-\frac{\delta^{a b} x^{2}}{12}\left\langle\bar{q} g_{s} \sigma \cdot G q\right\rangle -\frac{m_{q} \delta^{a b}}{4 \pi^{2} x^{2}} \\ &+\frac{{\rm i} \delta^{a b} m_{q}(\bar{q} q)}{48} \hat{x} -\frac{{\rm i} m_{q}\left\langle\bar{q} g_{s} \sigma \cdot G q\right) \delta^{a b} x^{2} \hat{x}}{1152}\, , \\ {\rm i} S_{Q}^{a b}(p) = & \frac{{\rm i} \delta^{a b}}{\hat{p}-m_{Q}} +\frac{{\rm i}}{4} g_{s} \frac{\lambda_{a b}^{n}}{2} G_{\mu \nu}^{n} \frac{\sigma^{\mu \nu}\left(\hat{p}+m_{Q}\right)+\left(\hat{p}+m_{Q}\right) \sigma^{\mu \nu}}{12} \\ &+\frac{{\rm i} \delta^{a b}}{12}\left\langle g_{s}^{2} G G\right\rangle m_{Q} \frac{p^{2}+m_{Q} \hat{p}}{(p^{2}-m_{Q}^{2})^{4}}\, , \end{aligned}

      (13)

      where q is the u , d , or s quark, and Q represents the c or b quark. The superscripts a, b denote the color indices, and \hat{x} = x^{\mu}\gamma_{\mu},\; \hat{p} = p^{\mu}\gamma_{\mu} . In this work, we calculate the Wilson coefficients up to dimension eight condensates at the leading order in \alpha_s . In Ref. [43], the NLO perturbative corrections to the correlation functions for the sc\bar q\bar c tetraquark systems have been studied, and their results show that such contributions are numerically small. The spectral densities for the interpolating currents in Eqs. (2) and (3) are evaluated and listed in appendix A. The tetraquark currents \eta_{1}(x) , \eta_{2}(x) , \eta_{1\mu}(x) , and \eta_{2\mu}(x) are the same as \eta_{2}(x) , \eta_{4}(x) , \eta_{2\mu}(x) , and \eta_{4\mu}(x) for the sc\bar q\bar b systems in Ref. [44], by replacing the bottom quark with the charm quark b\to c . Thus, we do not list the spectral densities for these four tetraquark currents in appendix A. To improve the convergence of the OPE series and suppress the contributions from the continuum and higher states region, the Borel transformation is applied to the correlation function at both the hadron and the quark-gluon levels. The QCD sum rules are then established as

      {\cal{L}}_{k}\left(s_{0}, M_{\rm B}^{2}\right) = f_{H}^{2} m_{H}^{2 k} {\rm e}^{-m_{H}^{2} / M_{\rm B}^{2}} = \int_{4m_{c}^{2}}^{s_{0}} {\rm d} s {\rm e}^{-s / M_{\rm B}^{2}} \rho(s) s^{k}\, ,

      (14)

      in which M_{\rm B} represents the Borel mass introduced by the Borel transformation, and s_0 is the continuum threshold. The mass of the lowest-lying hadron can be thus extracted as

      \begin{array}{l} m_{H}\left(s_{0}, M_{\rm B}^{2}\right) = \sqrt{\frac{{\cal{L}}_{1}\left(s_{0}, M_{\rm B}^{2}\right)}{{\cal{L}}_{0}\left(s_{0}, M_{\rm B}^{2}\right)}}\, , \end{array}

      (15)

      which is the function of the two parameters M_{\rm B}^2 and s_0 . We shall discuss in detail how to obtain suitable parameter working regions in QCD sum rule analyses in the next section.

    IV.   NUMERICAL ANALYSIS
    • In this section, we perform the QCD sum rule analyses for the \bar{D}_s^{(*)}D^{(*)} molecular and sc\bar q\bar c tetraquark systems using the interpolating currents in Eqs. (2) and (3). We use the values of quark masses and various QCD condensates as follows [45-53]

      \begin{aligned}[b] m_{u}(2 \;{\rm{GeV}}) = &\, (2.2_{-0.4}^{+0.5} ) \;{\rm{MeV}}\ , \\ m_{d}(2 \;{\rm{GeV}}) = &\, (4.7_{-0.3}^{+0.5}) \;{\rm{MeV}}\, ,\\ m_{q}(2\; {\rm{GeV}}) = &\, (3.5_{-0.2}^{+0.5}) \;{\rm{MeV}}\, ,\\ m_{s}(2 \;{\rm{GeV}}) = &\, (95_{-3}^{+9}) \;{\rm{MeV}}\, ,\\ m_{c}\left(m_{c}\right) = &\, (1.275 _{-0.035}^{+0.025}) \;{\rm{GeV}}\, , \\ m_{b}\left(m_{b}\right) = &\, (4.18 _{-0.03}^{+0.04}) \;{\rm{GeV}}\, , \\ \langle\bar{q} q\rangle = &\, -(0.24 \pm 0.03)^{3} \;{\rm{GeV}}^{3}\, , \\ \left\langle\bar{q} g_{s} \sigma \cdot G q\right\rangle = &\, - M_{0}^{2}\langle\bar{q} q\rangle\, ,\\ M_{0}^{2} = &\, (0.8 \pm 0.2) \;{\rm{GeV}}^{2}\, , \\ \langle\bar{s} s\rangle /\langle\bar{q} q\rangle = &\, 0.8 \pm 0.1\, , \\ \left\langle g_{s}^{2} G G\right\rangle = &\, (0.48\pm0.14) \;{\rm{GeV}}^{4}\, , \end{aligned}

      (16)

      where the u,d,s quark masses are the current quark masses obtained in the \overline{MS} scheme at the scale \mu = 2 GeV. We use the running mass in the \overline{MS} scheme for the charm quark, which is different from the value of the pole quark mass. Various reports show that the use of the \overline{MS} mass of the charm quark can lead to very good predictions for the masses of XYZ states in the framework of QCD sum rules [15, 54].

      To establish a stable mass sum rule, one should find appropriate parameter working regions first, i.e, the continuum threshold s_{0} and the Borel mass M_{\rm B}^{2} . The threshold s_{0} can be determined via the minimized variation of the hadronic mass m_{H} with the Borel mass M_{\rm B}^{2} . The lower bound on the Borel mass M_{\rm B}^{2} can be fixed by requiring a reasonable OPE convergence, while its upper bound is determined through a sufficient pole contribution. The pole contribution is defined as

      {\rm{PC}}\left(s_{0}, M_{\rm B}^{2}\right) = \frac{{\cal{L}}_{0}\left(s_{0}, M_{\rm B}^{2}\right)}{{\cal{L}}_{0}\left(\infty, M_{\rm B}^{2}\right)}\, ,

      (17)

      where {\cal{L}}_{0} has been defined in Eq. (14).

      We use the \bar{D}_s^\ast D^\ast molecular current J_{2}(x) with J^{P} = 0^+ as an example to show the details of the numerical analysis. For this current, the dominant non-perturbative contribution to the correlation function comes from the quark condensate \langle\bar{q}q\rangle and \langle\bar{s}s\rangle . In Fig. 1, we show the contributions of the perturbative term and various condensate terms to the correlation function. It is clear that the Borel mass M_{\rm B}^{2} should be large enough to ensure the convergence of the OPE series. Here, we require the highest dimension condensate contribution to be less than 10%,

      Figure 1.  (color online) OPE convergence for the \bar{D}_s^\ast D^\ast molecular current J_{2}(x) with J^{P} = 0^+ .

      {\frac{\Pi^{\langle\bar{q}q\rangle \langle\bar{q}g_{s}\sigma\cdot G q\rangle}(M_{\rm B}^{2},\infty)}{\Pi(M_{\rm B}^{2},\infty)}<10\% } \, ,

      (18)

      which results in M_{\rm B}^{2}\geqslant 2.6\;\text{GeV}^{2}.

      As mentioned above, the variation of the output hadron mass m_{H} with M_{\rm B}^{2} should be minimized to obtain the optimized value of the continuum threshold s_0 . We show the variations of m_{H} with s_{0} and M_{\rm B}^{2} in Fig. 2, from which the dependence of m_{H} on M_{\rm B}^{2} can be minimized at s_{0}\approx 20.5\;\text{GeV}^{2} . Requiring the pole contribution to be larger than 30%, the upper bound on M_{\rm B}^{2} can then be determined to be 3.4\; \text{GeV}^{2} . The working region of the Borel parameter for the scalar \bar{D}_s^\ast D^\ast molecular current J_{2}(x) is thus 2.6\leqslant M_{\rm B}^{2}\leqslant 3.4\;\text{GeV}^{2}. As shown in Fig. 2, the mass sum rules are established to be very stable in these parameter regions, and the hadron mass for the \bar{D}_s^\ast D^\ast molecule with J^{P} = 0^{+} can be obtained as

      Figure 2.  (color online) Variations of m_{H} with s_{0} and M_{\rm B}^{2} corresponding to the current J_{2}(x) in the \bar{D}_s^\ast D^\ast system with J^{P} = 0^{+}.

      m_{\bar{D}_s^\ast D^\ast, \, 0^+} = 4.11\pm0.14\; \text{GeV}\, ,

      (19)

      in which the error comes from the uncertainties of the continuum threshold s_{0} , Borel mass M_{\rm B} , the various condensates, and quark masses. After performing similar analyses, we obtain the numerical results for all the other interpolating currents in Eqs. (2) and (3) and present them in Table 1.

      System Current J^{P} s_{0} /\text{GeV} ^{2} M_{B}^{2} / \text{GeV}^{2} m_{H} /GeV PC (%)
      \bar{D}_sD J_{1} 0^{+} 18.0 ± 2.0 1.6 ~ 3.6 3.74 ± 0.13 52.5
      \bar{D}_s^\ast D^\ast J_{2} 0^{+} 20.5 ± 2.0 2.6 ~ 3.4 4.11 ± 0.14 42.4
      \bar{D}_s^{*}D J_{1\mu} 1^{+} 20.7 ± 2.0 2.1 ~ 2.5 3.99 ± 0.12 68.2
      \bar{D}_sD^{*} J_{2\mu} 1^{+} 20.5 ± 2.0 2.1 ~ 2.5 3.97 ± 0.11 67.7
      \bar{D}_s^\ast D^\ast J_{3\mu} 1^{+} 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.1
      \bar{D}_s^\ast D^\ast J_{4\mu} 1^{+} 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.0
      \bar{D}_s^\ast D^\ast J_{\mu\nu} 2^{+} 23.0 ± 2.0 2.8 ~ 4.3 4.34 ± 0.13 48.7
      \mathbf{0}_{[sc]} \oplus \mathbf{0}_{[\bar q \bar{c}]} (spin-spin) \eta_{1} 0^{+} 18.0 ± 2.0 2.1 ~ 3.1 3.84 ± 0.15 46.3
      \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} \eta_{2} 0^{+} 20.0 ± 2.0 2.6 ~ 3.2 4.13 ± 0.17 35.6
      \mathbf{1}_{[sc]} \oplus \mathbf{0}_{[\bar q \bar{c}]} \eta_{1\mu} 1^{+} 19.0 ± 2.0 2.5 ~ 3.3 3.98 ± 0.16 41.0
      \mathbf{0}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} \eta_{2\mu} 1^{+} 19.0 ± 2.0 2.5 ~ 3.3 3.97 ± 0.15 41.6
      \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} \eta_{3\mu} 1^{+} 22.0 ± 2.0 2.9 ~ 3.6 4.28 ± 0.14 40.9
      \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} \eta_{4\mu} 1^{+} 22.0 ± 2.0 2.9 ~ 3.6 4.28 ± 0.14 41.1
      \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} \eta_{\mu\nu} 2^{+} 23.0 ± 2.0 2.8 ~ 4.3 4.33 ± 0.13 46.4

      Table 1.  Numerical results for the \bar{D}_s^{(\ast)} D^{(\ast)} molecular and diquark-antiquark sc\bar q\bar c tetraquark systems.

      In Table 1, the mass of the scalar \bar{D}_sD molecular state is predicted to be slightly below the open-charm threshold T_{\bar{D}_sD} = 3.84 GeV, implying that it can only decay into the hidden-charm channel \eta_c K . The scalar \bar{D}_s^{*} D^{*} state is predicted to be very close to T_{\bar{D}_s^{*} D^{*}} = 4.12 GeV; however, it can decay into \bar{D}_sD and \eta_c K final states kinematically in the S-wave. The masses for the \bar{D}_{s}^{*} D^{*} molecular states with J^P = 1^{+}, 2^{+} are significantly above the corresponding open-charm thresholds.

      The masses obtained from the axial-vector molecular currents J_{1\mu} and J_{2\mu} are m_{\bar{D}_s^{*}D, \, 1^+} = (3.99 \pm 0.12) GeV and m_{\bar{D}_{s}D^{*}, \, 1^+} = (3.97 \pm 0.11) GeV, which are almost degenerate with each other. One may wonder whether these two currents J_{1\mu} and J_{2\mu} could couple to the same physical molecular state or not. In QCD sum rules, this can be specified by studying the following off-diagonal correlation function

      \begin{array}{l} \Pi_{12\mu \nu}^M\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{1\mu}(x) J_{2\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array}

      (20)

      Our calculation shows that this off-diagonal correlation function \Pi_{12\mu \nu}^M\left(p^{2}\right) = 0 at the leading order of \alpha_s for the axial-vector molecular currents J_{1\mu} and J_{2\mu} , including the perturbative term and all contributions from various non-perturbative condensates. According to Ref. [43], the NLO perturbative correction is numerically small; thus, \Pi_{12\mu \nu}^M\left(p^{2}\right) is still negligible compared with the diagonal correlators \Pi_{11\mu \nu}^M\left(p^{2}\right) and \Pi_{22\mu \nu}^M\left(p^{2}\right) at the next leading order of \alpha_s . This result implies that J_{1\mu} and J_{2\mu} may couple to different physical states.

      We also study the sc\bar q\bar c tetraquark systems with J^P = 0^+, 1^+, 2^+ . In Fig. 3, we show the variations of the tetraquark mass with s_{0} and M_{\rm B}^{2} for the current \eta_{1\mu}(x) with J^{P} = 1^{+} ; the mass sum rules are very stable and reliable in the chosen parameter regions. Regarding the interpolating currents in Eq. (3), we collect the numerical results for these sc\bar q\bar c tetraquark systems in Table 1. It is shown that the mass spectra for the sc\bar q\bar c tetraquarks are very similar to the \bar{D}_s^{(\ast)} D^{(\ast)} molecular states. For the axial-vector sc\bar q\bar c tetraquark systems, the extracted masses from \eta_{1\mu}(x) and \eta_{2\mu}(x) are almost the same as the \bar{D}_s^{*}D and \bar{D}_sD^{*} molecular states, which are consistent with the mass of Z_{cs}(3985)^- from BESIII [1]. It is interesting to examine the off-diagonal correlation function for \eta_{1\mu}(x) and \eta_{2\mu}(x)

      Figure 3.  (color online) Variations of m_{H} with s_{0} and M_{\rm B}^{2} for the current \eta_{1\mu}(x) in the sc\bar q\bar c tetraquark system with J^{P} = 1^{+}.

      \begin{array}{l} \Pi_{12\mu \nu}^T\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{1\mu}(x) J_{2\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array}

      (21)

      The calculation indicates that the perturbative term and the quark condensate terms in \Pi_{12\mu \nu}^T\left(p^{2}\right) are equal to zero, This off-diagonal correlation function \Pi_{12\mu \nu}^T\left(p^{2}\right) is very small, suggesting that the currents \eta_{1\mu}(x) and \eta_{2\mu}(x) cannot strongly couple to the same physical state.

    V.   CONCLUSION
    • To study the hidden-charm four-quark systems with strangeness, we have calculated the mass spectra for the \bar{D}_s^{(*)}D^{(*)} molecular states and sc\bar q\bar c tetraquark states with J^P = 0^+, 1^+, 2^+ in the framework of QCD sum rules. We construct the corresponding molecular and tetraquark interpolating currents and calculate their two-point correlation functions and spectral densities up to dimension eight condensates at the leading order of \alpha_s . The quark condensates are found to be the most important non-perturbative contribution to the correlation functions for both molecular and tetraquark systems.

      One may wonder if the two-meson scattering states can contribute to the correlation functions in our calculations. In general, the interpolating currents can couple to all structures with the same quantum numbers, including resonances, two-meson scattering states, and continuum. Thus, these structures will give contributions to the correlation functions. However, it has been demonstrated that the two-meson scattering states cannot saturate the QCD sum rules, while only exotic four-quark states can saturate the QCD sum rules. Moreover, the contributions from the two-meson scattering states to the correlation functions are numerically negligible [43, 55].

      Our results show that the masses of the axial-vector \bar{D}_sD^{*} , \bar{D}_s^{*}D molecular states and the sc\bar q\bar c tetraquark states from \eta_{1\mu} , \eta_{2\mu} are calculated in good agreement with the mass of Z_{cs}(3985)^- . In the present calculations, it is difficult to distinguish the nature of Z_{cs}(3985)^- from the molecular and diquark-antidiquark configurations. In both the molecular and diquark-antidiquark pictures, our results suggest that there may exist two almost degenerate states, as the strange partners of X(3872) and Z_c(3900) . We propose to carefully examine Z_{cs}(3985) in future experiments to verify this. One can search for more hidden-charm four-quark states with strangeness in not only the open-charm \bar{D}_s^{(*)}D^{(*)} channels but also the hidden-charm channels \eta_c K/K^\ast , J/\psi K/K^\ast .

      Note added: Since we finished this work, the LHCb Collaboration has reported two new charged resonances, Z_{cs}(4000)^+ and Z_{cs}(4220)^+ , in the J/\psi K^+ final states [56]. Their masses and decay widths are measured to be M_{Z_{cs}(4000)^+} = 4003\pm6^{+4}_{-14} MeV and \Gamma_{Z_{cs}(4000)^+} = 131\pm15\pm26 MeV, and M_{Z_{cs}(4220)^+} = 4216\pm24^{+43}_{-30} MeV and \Gamma_{Z_{cs}(4220)^+} = 233\pm52^{+97}_{-73} MeV, respectively, while their spin-parity quantum numbers are identified to prefer J^P = 1^+ . These masses and spin-parity are consistent with the axial-vector \bar{D}_sD^{*} ( \bar{D}_s^{*}D ), \bar{D}^{*}_sD^{*} molecular states and \mathbf{1}_{[sc]} \oplus \mathbf{0}_{[\bar q \bar{c}]} ( \mathbf{0}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} ), \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} ( \mathbf{1}_{[sc]} \oplus \mathbf{1}_{[\bar q \bar{c}]} ) tetraquark states that we have predicted in Table 1.

      According to the observation of the LHCb, the decay width of Z_{cs}(4000) is much larger than that of Z_{cs}(3985) observed by BESIII [1]. LHCb found no evidence that Z_{cs}(4000) and Z_{cs}(3985) are the same state, although their masses are very close to each other. If this is true, they may be identified as the strange partners of X(3872) and Z_c(3900) with J^{PC} = 1^{++} and J^{PC} = 1^{+-} , respectively. We propose to carefully examine Z_{cs}(4000) and Z_{cs}(3985) in future experiments to verify this.

    APPENDIX A: THE SPECTRAL DENSITIES
    • In this appendix, we list the spectral densities for the \bar{D}_s^{(*)}D^{(*)} and sc\bar q\bar c systems with J^{P} = 0^{+} , 1^{+} , and 2^{+} . The spectral density includes the perturbative term, quark condensate, gluon condensate, quark-gluon mixed condensate, four-quark condensate, and dimension eight condensate

      \rho(s) = \rho^{0}(s)+\rho^{3}(s)+\rho^{4}(s)+\rho^{5}(s)+\rho^{6}(s)+\rho^{8}(s)\, , \tag{A1}

      (22)

      in which the superscripts stand for the dimension of various condensates.

      1. Spectral densities for J_{1} :

      \rho_{J_{1}}^{0a}(s) = \frac{3} {2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\, ,

      \rho_{J_{1}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

      \rho_{J_{1}}^{3a}(s) \!=\! - \frac{3\langle\bar{s}s\rangle}{128\pi^{4} } \!\!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \!\!\int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{2m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{1}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{128\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{2m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{1}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (2m_{c}^{2}(\alpha+ \beta)-3 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{J_{1}}^{4b}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)(m_{c}^{2}(\alpha+ \beta)-2\alpha \beta s)\left(\frac{1}{\alpha^{2}\beta}+\frac{1}{\alpha\beta^{2}}\right)\, ,

      \rho_{J_{1}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta) \left(\frac{1}{\beta}-\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)+\frac{2m_{c}m_{q}}{\beta}\right]\, ,

      \rho_{J_{1}}^{5b}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta) \left(\frac{1}{\alpha}-\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)+\frac{2m_{c}m_{s}}{\alpha}\right]\, ,

      \rho_{J_{1}}^{5c}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{512 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-6 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{512 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-6 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{1}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{32 \pi^{2}}(2m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{1}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{1}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}-\frac{ 2\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{\alpha m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      where

      \alpha_{\rm min } = \frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{4 m_{c}^{2}}{s}},\; \; \; \alpha_{\rm max } = \frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{4 m_{c}^{2}}{s}},\; \; \; \beta_{\rm min} = \frac{\alpha m_{c}^{2}}{\alpha s-m_{c}^{2}},\; \; \; \; \beta_{\rm max} = 1-\alpha \, ,

      2. Spectral densities for J_{2} :

      \rho_{J_{2}}^{0a}(s) = \frac{3} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\, ,

      \rho_{J_{2}}^{0b}(s) = - \frac{3m_{c}} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

      \rho_{J_{2}}^{3a}(s) = - \frac{3\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)+2\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{2}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)+2\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{2}}^{4}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (2m_{c}^{2}(\alpha+ \beta)-3 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{J_{2}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta}{\beta} +\frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta }{\alpha}\, ,

      \rho_{J_{2}}^{5b}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{128 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-6 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{128 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-6 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{2}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{2}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{16 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{2}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{16 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      3. Spectral densities for J_{1\mu} :

      \rho_{J_{1\mu}}^{0a}(s) = \frac{3} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{J_{1\mu}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}+\frac{(m_{c}^{2}(\alpha+\beta)-4\alpha \beta s)m_{q}}{\alpha^{2} \beta^{3}}\right)\, ,

      \rho_{J_{1\mu}}^{3a}(s) = - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{1\mu}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{1\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\Big(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\Big)\, ,

      \rho_{J_{1\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha\beta^{2}} -\frac{(3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s)}{\alpha^{2}\beta}\right)\, ,

      \rho_{J_{1\mu}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\alpha}\, ,

      \rho_{J_{1\mu}}^{5b}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\beta}-\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)+\frac{2m_{c}m_{s}}{\beta}\right]\, ,

      \rho_{J_{1\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{1\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{1\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{1\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      4. Spectral densities for J_{2\mu} :

      \rho_{J_{2\mu}}^{0a}(s) = \frac{3} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{J_{2\mu}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \Big(\frac{(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)m_{q}}{\alpha^{2} \beta^{3}} +\frac{(m_{c}^{2}(\alpha+\beta)-4\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}\Big)\, ,

      \rho_{J_{2\mu}}^{3a}(s) = - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha\beta^{2} } -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{2\mu}}^{3b}(s) = - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha ^{2} \beta} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{2\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{J_{2\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2} \beta} -\frac{(3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s)}{\alpha\beta^{2} }\right)\, ,

      \rho_{J_{2\mu}}^{5a}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\beta}\, ,

      \rho_{J_{2\mu}}^{5b}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\alpha}-\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)+\frac{2m_{c}m_{q}}{\alpha}\right]\, ,

      \rho_{J_{2\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{2\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{s}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{2\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{2\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      5. Spectral densities for J_{3\mu} :

      \rho_{J_{3\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{J_{3\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, ,

      \rho_{J_{3\mu}}^{3a}(s) = \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{3\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \!\!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}}\! {\rm d} \alpha \!\!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1\!-\!\alpha-\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta}\!-\!\frac{12m_{c}^{2}m_{s}\! +\! 3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{3\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{J_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha\beta^{2}} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2}\beta}\right)\, ,

      \rho_{J_{3\mu}}^{5a}(s) = -\frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \alpha \, ,

      \rho_{J_{3\mu}}^{5b}(s) = \frac{ \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\alpha}+\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)-\frac{6m_{c}m_{s}}{\alpha}\right]\, ,

      \rho_{J_{3\mu}}^{5c}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{3\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      6. Spectral densities for J_{4\mu} :

      \rho_{J_{4\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{J_{4\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{2} \beta^{3}}-\frac{ \alpha \beta sm_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

      \rho_{J_{4\mu}}^{3a}(s) = \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \alpha}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{J_{4\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} \!{\rm d} \alpha \!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\! \beta)\! -\! \alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\! +\! \beta)\! -\! 5\alpha \beta s)m_{c}} { \alpha \beta^{2}} \! -\! \frac{12m_{c}^{2}m_{q}\! +\! 3(m_{c}^{2}(\alpha\! +\! \beta)\! -\! 3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{J_{4\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{J_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha^{2}\beta} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha\beta^{2}}\right)\, ,

      \rho_{J_{4\mu}}^{5a}(s) = -\frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \beta \, ,

      \rho_{J_{4\mu}}^{5b}(s) = \frac{ \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\beta}+\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)-\frac{6m_{c}m_{q}}{\beta}\right]\, ,

      \rho_{J_{4\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}+\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{4\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      7. Spectral densities for J_{\mu\nu} :

      \rho_{J_{\mu\nu}}^{0a}(s) = -\frac{5} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\,

      \rho_{J_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

      \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3a}(s) = &- \frac{15\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}} \right. \\ &\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, ,\end{aligned}

      \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3b}(s) = & - \frac{15\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left. -\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned}

      \rho_{J_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, ,

      \begin{aligned}[b] \rho_{J_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{1}{\alpha^{2}\beta}+\frac{1}{\alpha\beta^{2}}\right)\\ &\times\left((1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)-4(m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\right)\, , \end{aligned}

      \begin{aligned}[b] \rho_{J_{\mu\nu}}^{5a}(s) = &\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned}

      \rho_{J_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{J_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{32 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{J_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{16 \pi^{2}} \int_{0}^{1} {\rm{d}} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{J_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      8. Spectral densities for \eta_{3\mu} :

      \rho_{\eta_{3\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{\eta_{3\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{q}}{\alpha^{2} \beta^{3}}\right)\, ,

      \rho_{\eta_{3\mu}}^{3a}(s) = \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{\eta_{3\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1-\alpha\!-\!\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{q}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{\eta_{3\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{\eta_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, ,

      \rho_{\eta_{3\mu}}^{5a}(s) = \frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, ,

      \rho_{\eta_{3\mu}}^{5b}(s) = -\frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, ,

      \rho_{\eta_{3\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{\eta_{3\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{\eta_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{\eta_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+2\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      9. Spectral densities for \eta_{4\mu} :

      \rho_{J_{4\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, ,

      \rho_{\eta_{4\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, ,

      \rho_{\eta_{4\mu}}^{3a}(s) = \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,

      \rho_{\eta_{4\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\! + \! \beta)\! - \!\alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{s}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,

      \rho_{\eta_{4\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, ,

      \rho_{\eta_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, ,

      \ \rho_{\eta_{4\mu}}^{5a}(s) = \frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, ,

      \rho_{\eta_{4\mu}}^{5b}(s) = -\frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, ,

      \rho_{\eta_{4\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{\eta_{4\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{\eta_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{\eta_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      10. Spectral densities for \eta_{\mu\nu} :

      \rho_{\eta_{\mu\nu}}^{0a}(s) = -\frac{5} {768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\,

      \rho_{\eta_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {384 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, ,

      \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3a}(s) = & - \frac{5\langle\bar{s}s\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha^{2} \beta}\right.\\ &\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, , \end{aligned}

      \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3b}(s) = & - \frac{5\langle\bar{q}q\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left.-\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned}

      \rho_{\eta_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, ,

      \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{12288 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left[(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)\left(1+\frac{2(1-\alpha-\beta)^{2}}{\alpha\beta}\right) \right.\\ &\left.+\frac{4(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)(1-\alpha-\beta)(\alpha+\beta)}{\alpha\beta^{2}}\right]\, , \end{aligned}

      \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{5a}(s) =&\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned}

      \rho_{\eta_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \rho_{\eta_{\mu\nu}}^{5c}(s) = \frac{5 \left(\langle\bar{s} g_{s}\sigma\cdot G s\rangle+\langle\bar{q} g_{s}\sigma\cdot G q\rangle\right) m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{7(m_{c}^{2}(\alpha+\beta)-6 \alpha \beta s)(\alpha+5(1-\alpha+\beta))}{\alpha\beta}\, ,

      \rho_{\eta_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{24 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,

      \Pi_{\eta_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{12 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

      \Pi_{\eta_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \left[\frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}}-\frac{\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{12\alpha}\right] {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, ,

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