-
Very recently, the BESIII Collaboration announced a new structure near the
D−sD∗0 andD∗−sD0 thresholds in theK+ recoil-mass spectra ine+e−→K+(D−sD∗0+D∗−sD0) [1]. The pole mass and width of thisZcs(3985)− resonance were measured as(3982.5+1.8−2.6±2.1)MeV and(12.8+5.3−4.4±3.0)MeV , respectively. Decaying intoD−sD∗0 andD∗−sD0 in the S-wave, the spin-parity ofZcs(3985)− is assumed to favorJP=1+ and the quark contentcˉcsˉu [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 ofˉD(∗)D(∗)s . In Ref. [12], an axial-vector hidden-charmD∗−D+s−D−D∗+s molecular state was also predicted to exist. PossibleDˉD∗s0(2317) andD∗ˉD∗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
Zcs(3985)− lies very close to the mass thresholds ofD−sD∗0 andD∗−sD0 , it is naturally studied in a molecular picture [20-27], as a partner state ofZc(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 mesonD∗s2(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 theDsˉD∗/D∗sˉD 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 ofZcs(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 onZcs(3985) using other methods. In this work, we shall study the exoticˉD(∗)sD(∗) molecular states andscˉqˉc tetraquark states withJP=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 andscˉqˉc tetraquark systems withJP=0+,1+ , and2+ . In Sec. III, we calculate the correlation functions and spectral densities for these interpolating currents. We extract the masses for theˉD(∗)sD(∗) molecular states andscˉqˉc 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]=(1⊕8)[qˉQ]⊗(1⊕8)[Qˉq]=(1⊗1)⊕(1⊗8)⊕(8⊗1)⊕(8⊗8)=1⊕8⊕8⊕(1⊕8⊕8⊕10⊕¯10⊕27),(3⊗3)[qQ]⊗(¯3⊗¯3)[ˉQˉq]=(6⊕¯3)[qQ]⊗(3⊕¯6)[ˉQˉq]=(6⊗¯6)⊕(¯3⊗3)⊕(6⊗3)⊕(¯3⊗¯6)=(1⊕8⊕27)⊕(1⊕8)⊕(8⊕10)⊕(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 momentumL=0 between two mesonic fields and also between two diquark fields. Finally, we obtain theˉD(∗)sD(∗) molecular interpolating currents asJ1=(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 asη1=sTaCγ5cb(ˉqaγ5CˉcTb−ˉqbγ5CˉcTa),JP=0+,η2=sTaCγμcb(ˉqaγμCˉcTb−ˉqbγμCˉcTa),JP=0+,η1μ=sTaCγμcb(ˉqaγ5CˉcTb−ˉqbγ5CˉcTa),JP=1+,η2μ=sTaCγ5cb(ˉqaγμCˉcTb−ˉqbγμCˉcTa),JP=1+,η3μ=sTaCγαcb(ˉqaσαμγ5CˉcTb−ˉqbσαμγ5CˉcTa),JP=1+,η4μ=sTaCσαμγ5cb(ˉqaγαCˉcTb−ˉqbγαCˉcTa),JP=1+,ημν=sTaCγμcb(ˉqaγνCˉcTb−ˉqbγνCˉcTa),JP=2+,
(3) in which
a ,b denote color indices andq is an up or down quark. The mesonic fieldˉqaσαμγ5qa inJ3μ andJ4μ can couple to both the vector channelJP=1− (ˉqaσijγ5qa ) and axial-vector channelJP=1+ (ˉqaσ0iγ5qa ). We pick out its S-wave vector component by multiplicating a vector mesonic fieldˉqγαq , so that the molecular operators carry positive parity. A similar situation occurs for the tetraquark currentsη3μ andη4μ . 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 theˉD(∗)sD(∗) molecular states andscˉqˉc 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)=i∫d4xeip⋅x⟨0|T[J(x)J†(0)]|0⟩,
(4) and that for the axial-vector current is
Πμν(p2)=i∫d4xeip⋅x⟨0|T[Jμ(x)J†ν(0)]|0⟩.
(5) The correlation function
Πμν(p2) in Eq. (5) can be rewitten asΠμν(p2)=(pμpνp2−gμν)Π1(p2)+pμpνp2Π0(p2),
(6) where
Π0(p2) andΠ1(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 currentJμν(x) isΠμν,ρσ(p2)=i∫d4xeip⋅x⟨0|T[Jμν(x)J†ρσ(0)]|0⟩,
(7) which can be expressed as
Πμν,ρσ(p2)=(ημρηνσ+ημσηνρ−23ημνηρσ)Π2(p2)+⋯,
(8) where
ημν=pμpνp2−gμν,
(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)Nπ∫∞4m2cImΠ(s)sN(s−p2−iϵ)ds+N−1∑n=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 functionρ(s)=1πImΠ(s)=f2Hδ(s−m2H)+QCD continuum and higher states,
(11) in which the “pole plus continuum parametrization” is used. The parameters
fH andmH are the coupling constant and mass of the lowest-lying hadronic resonanceH , respectively⟨0|J|H⟩=fH,⟨0|Jμ|H⟩=fHϵμ,⟨0|Jμν|H⟩=fHϵμν
(12) with the polarization vector
ϵμ and polarization tensorϵμν .We can calculate the correlation function
Π(p2) and spectral densityρ(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 spaceiSabq(x)=iδab2π2x4ˆx+i32π2λnab2gsGnμν1x2(σμνˆx+ˆxσμν)−δabx212⟨ˉqgsσ⋅Gq⟩−mqδab4π2x2+iδabmq(ˉqq)48ˆx−imq⟨ˉqgsσ⋅Gq)δabx2ˆx1152,iSabQ(p)=iδabˆp−mQ+i4gsλnab2Gnμνσμν(ˆp+mQ)+(ˆp+mQ)σμν12+iδab12⟨g2sGG⟩mQp2+mQˆp(p2−m2Q)4,
(13) where
q is theu ,d , ors quark, andQ represents thec orb quark. The superscriptsa,b denote the color indices, andˆx=xμγμ,ˆp=pμγμ . In this work, we calculate the Wilson coefficients up to dimension eight condensates at the leading order inαs . In Ref. [43], the NLO perturbative corrections to the correlation functions for thescˉqˉc 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 currentsη1(x) ,η2(x) ,η1μ(x) , andη2μ(x) are the same asη2(x) ,η4(x) ,η2μ(x) , andη4μ(x) for thescˉqˉb systems in Ref. [44], by replacing the bottom quark with the charm quarkb→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 asLk(s0,M2B)=f2Hm2kHe−m2H/M2B=∫s04m2cdse−s/M2Bρ(s)sk,
(14) in which
MB represents the Borel mass introduced by the Borel transformation, ands0 is the continuum threshold. The mass of the lowest-lying hadron can be thus extracted asmH(s0,M2B)=√L1(s0,M2B)L0(s0,M2B),
(15) which is the function of the two parameters
M2B ands0 . 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 andscˉqˉc 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+0.5−0.4)MeV ,md(2GeV)=(4.7+0.5−0.3)MeV,mq(2GeV)=(3.5+0.5−0.2)MeV,ms(2GeV)=(95+9−3)MeV,mc(mc)=(1.275+0.025−0.035)GeV,mb(mb)=(4.18+0.04−0.03)GeV,⟨ˉqq⟩=−(0.24±0.03)3GeV3,⟨ˉqgsσ⋅Gq⟩=−M20⟨ˉqq⟩,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 scaleμ=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 massM2B . The thresholds0 can be determined via the minimized variation of the hadronic massmH with the Borel massM2B . The lower bound on the Borel massM2B 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 asPC(s0,M2B)=L0(s0,M2B)L0(∞,M2B),
(17) where
L0 has been defined in Eq. (14).We use the
ˉD∗sD∗ molecular currentJ2(x) withJP=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 massM2B 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%,Π⟨ˉqq⟩⟨ˉqgsσ⋅Gq⟩(M2B,∞)Π(M2B,∞)<10%,
(18) which results in
M2B⩾2.6GeV2 .As mentioned above, the variation of the output hadron mass
mH withM2B should be minimized to obtain the optimized value of the continuum thresholds0 . We show the variations ofmH withs0 andM2B in Fig. 2, from which the dependence ofmH onM2B can be minimized ats0≈20.5GeV2 . Requiring the pole contribution to be larger than 30%, the upper bound onM2B can then be determined to be3.4GeV2 . The working region of the Borel parameter for the scalarˉD∗sD∗ molecular currentJ2(x) is thus2.6⩽M2B⩽3.4GeV2 . As shown in Fig. 2, the mass sum rules are established to be very stable in these parameter regions, and the hadron mass for theˉD∗sD∗ molecule withJP=0+ can be obtained asFigure 2. (color online) Variations of
mH withs0 andM2B corresponding to the currentJ2(x) in theˉD∗sD∗ system withJP=0+. mˉD∗sD∗,0+=4.11±0.14GeV,
(19) in which the error comes from the uncertainties of the continuum threshold
s0 , Borel massMB , 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 JP s0 /GeV2 M2B /GeV2 mH /GeVPC (%) ˉDsD J1 0+ 18.0 ± 2.0 1.6 ~ 3.6 3.74 ± 0.13 52.5 ˉD∗sD∗ J2 0+ 20.5 ± 2.0 2.6 ~ 3.4 4.11 ± 0.14 42.4 ˉD∗sD 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 ˉD∗sD∗ J3μ 1+ 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.1 ˉD∗sD∗ J4μ 1+ 21.5 ± 2.0 2.8 ~ 3.6 4.22 ± 0.14 40.0 ˉD∗sD∗ 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 Table 1. Numerical results for the
ˉD(∗)sD(∗) molecular and diquark-antiquarkscˉqˉc tetraquark systems.In Table 1, the mass of the scalar
ˉDsD molecular state is predicted to be slightly below the open-charm thresholdTˉDsD=3.84 GeV, implying that it can only decay into the hidden-charm channelηcK . The scalarˉD∗sD∗ state is predicted to be very close toTˉD∗sD∗=4.12 GeV; however, it can decay intoˉDsD andηcK final states kinematically in the S-wave. The masses for theˉD∗sD∗ molecular states withJP=1+,2+ are significantly above the corresponding open-charm thresholds.The masses obtained from the axial-vector molecular currents
J1μ andJ2μ aremˉD∗sD,1+=(3.99±0.12) GeV andmˉDsD∗,1+=(3.97±0.11) GeV, which are almost degenerate with each other. One may wonder whether these two currentsJ1μ andJ2μ 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 functionΠM12μν(p2)=i∫d4xeip⋅x⟨0|T[J1μ(x)J†2ν(0)]|0⟩.
(20) Our calculation shows that this off-diagonal correlation function
ΠM12μν(p2)=0 at the leading order ofαs for the axial-vector molecular currentsJ1μ andJ2μ , 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 correlatorsΠM11μν(p2) andΠM22μν(p2) at the next leading order ofαs . This result implies thatJ1μ andJ2μ may couple to different physical states.We also study the
scˉqˉc tetraquark systems withJP=0+,1+,2+ . In Fig. 3, we show the variations of the tetraquark mass withs0 andM2B for the currentη1μ(x) withJP=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 thesescˉqˉc tetraquark systems in Table 1. It is shown that the mass spectra for thescˉqˉc tetraquarks are very similar to theˉD(∗)sD(∗) molecular states. For the axial-vectorscˉqˉc tetraquark systems, the extracted masses fromη1μ(x) andη2μ(x) are almost the same as theˉD∗sD andˉDsD∗ molecular states, which are consistent with the mass ofZcs(3985)− from BESIII [1]. It is interesting to examine the off-diagonal correlation function forη1μ(x) andη2μ(x) Figure 3. (color online) Variations of
mH withs0 andM2B for the currentη1μ(x) in thescˉqˉc tetraquark system withJP=1+. ΠT12μν(p2)=i∫d4xeip⋅x⟨0|T[J1μ(x)J†2ν(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 functionΠT12μν(p2) is very small, suggesting that the currentsη1μ(x) andη2μ(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 andscˉqˉc tetraquark states withJP=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 ofαs . 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∗ ,ˉD∗sD molecular states and thescˉqˉc tetraquark states fromη1μ ,η2μ are calculated in good agreement with the mass ofZcs(3985)− . In the present calculations, it is difficult to distinguish the nature ofZcs(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 ofX(3872) andZc(3900) . We propose to carefully examineZcs(3985) in future experiments to verify this. One can search for more hidden-charm four-quark states with strangeness in not only the open-charmˉD(∗)sD(∗) channels but also the hidden-charm channelsηcK/K∗ ,J/ψK/K∗ .Note added: Since we finished this work, the LHCb Collaboration has reported two new charged resonances,
Zcs(4000)+ andZcs(4220)+ , in theJ/ψK+ final states [56]. Their masses and decay widths are measured to beMZcs(4000)+=4003±6+4−14 MeV andΓZcs(4000)+=131±15±26 MeV, andMZcs(4220)+=4216±24+43−30 MeV andΓZcs(4220)+=233±52+97−73 MeV, respectively, while their spin-parity quantum numbers are identified to preferJP=1+ . These masses and spin-parity are consistent with the axial-vectorˉDsD∗ (ˉD∗sD ),ˉD∗sD∗ molecular states and1[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 ofZcs(3985) observed by BESIII [1]. LHCb found no evidence thatZcs(4000) andZcs(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 ofX(3872) andZc(3900) withJPC=1++ andJPC=1+− , respectively. We propose to carefully examineZcs(4000) andZcs(3985) in future experiments to verify this. -
In this appendix, we list the spectral densities for the
ˉD(∗)sD(∗) andscˉqˉc systems withJP=0+ ,1+ , and2+ . The spectral density includes the perturbative term, quark condensate, gluon condensate, quark-gluon mixed condensate, four-quark condensate, and dimension eight condensateρ(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)=32048π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2α3β3(m2c(α+β)−αβs)3(m2c(α+β)−3αβs), ρ0bJ1(s)=−3mc1024π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−αβs)2(2m2c(α+β)−5αβs)(msα2β3+mqα3β2),
ρ3aJ1(s)=−3⟨ˉss⟩128π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−2αβs)mcαβ2−2m2cmq+(m2c(α+β)−2αβs)msαβ],
ρ3bJ1(s)=−3⟨ˉqq⟩128π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−2αβs)mcα2β−2m2cms+(m2c(α+β)−2αβs)mqαβ],
ρ4aJ1(s)=⟨g2sGG⟩m2c4096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(2m2c(α+β)−3αβs)(1α3+1β3),
ρ4bJ1(s)=3⟨g2sGG⟩m2c2048π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)(m2c(α+β)−αβs)(m2c(α+β)−2αβs)(1α2β+1αβ2),
ρ5aJ1(s)=3⟨ˉsgsσ⋅Gs⟩mc256π4∫αmaxαmindα∫βmaxβmindβ[(2m2c(α+β)−3sαβ)(1β−2(1−α−β)β2)+2mcmqβ],
ρ5bJ1(s)=3⟨ˉqgsσ⋅Gq⟩mc256π4∫αmaxαmindα∫βmaxβmindβ[(2m2c(α+β)−3sαβ)(1α−2(1−α−β)α2)+2mcmsα],
ρ5cJ1(s)=⟨ˉqgsσ⋅Gq⟩512π4((s−2m2c)mq−6m2cms)√1−4m2cs+⟨ˉsgsσ⋅Gs⟩512π4((s−2m2c)ms−6m2cmq)√1−4m2cs,
ρ6aJ1(s)=⟨ˉss⟩⟨ˉqq⟩32π2(2m2c+mcmq+mcms)√1−4m2cs,
Π6bJ1(M2B)=−⟨ˉss⟩⟨ˉqq⟩m3c32π2∫10dα(mq1−α+msα)e−m2cα(1−α)M2B,
Π8J1(M2B)=m4c64π2∫10dα(⟨ˉss⟩⟨ˉqgsσ⋅Gq⟩+⟨ˉqq⟩⟨ˉsgsσ⋅Gs⟩(1−α)2M2B−2⟨ˉss⟩⟨ˉqgsσ⋅Gq⟩(1−α)m2c−2⟨ˉqq⟩⟨ˉsgsσ⋅Gs⟩αm2c)e−m2cα(1−α)M2B,
where
αmin=12−12√1−4m2cs,αmax=12+12√1−4m2cs,βmin=αm2cαs−m2c,βmax=1−α,
2. Spectral densities for
J2 :ρ0aJ2(s)=3512π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2α3β3(m2c(α+β)−αβs)3(m2c(α+β)−3αβs),
ρ0bJ2(s)=−3mc512π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−αβs)2(2m2c(α+β)−5αβs)(msα2β3+mqα3β2),
ρ3aJ2(s)=−3⟨ˉss⟩64π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−2αβs)mcαβ2−4m2cmq+(m2c(α+β)+2αβs)msαβ],
ρ3bJ2(s)=−3⟨ˉqq⟩64π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−2αβs)mcα2β−4m2cms+(m2c(α+β)+2αβs)mqαβ],
ρ4J2(s)=⟨g2sGG⟩m2c1024π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(2m2c(α+β)−3αβs)(1α3+1β3),
ρ5aJ2(s)=3⟨ˉsgsσ⋅Gs⟩mc128π4∫αmaxαmindα∫βmaxβmindβ2m2c(α+β)−3sαββ+3⟨ˉqgsσ⋅Gq⟩mc128π4∫αmaxαmindα∫βmaxβmindβ2m2c(α+β)−3sαβα,
ρ5bJ2(s)=⟨ˉqgsσ⋅Gq⟩128π4((s−2m2c)mq−6m2cms)√1−4m2cs+⟨ˉsgsσ⋅Gs⟩128π4((s−2m2c)ms−6m2cmq)√1−4m2cs,
ρ6aJ2(s)=⟨ˉss⟩⟨ˉqq⟩16π2(4m2c+mcmq+mcms)√1−4m2cs,
Π6bJ2(M2B)=⟨ˉss⟩⟨ˉqq⟩m3c16π2∫10dα(mq1−α+msα)e−m2cα(1−α)M2B,
Π8J2(M2B)=m4c16π2∫10dα⟨ˉss⟩⟨ˉqgsσ⋅Gq⟩+⟨ˉqq⟩⟨ˉsgsσ⋅Gs⟩(1−α)2M2Be−m2cα(1−α)M2B,
3. Spectral densities for
J1μ :ρ0aJ1μ(s)=34096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2α3β3(m2c(α+β)−αβs)3(m2c(α+β)−5αβs),
ρ0bJ1μ(s)=−3mc1024π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−αβs)2((2m2c(α+β)−5αβs)msα3β2+(m2c(α+β)−4αβs)mqα2β3),
ρ3aJ1μ(s)=−3⟨ˉss⟩256π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[4(1−α−β)(m2c(α+β)−2αβs)mcα2β−4m2cmq+(m2c(α+β)−3αβs)msαβ],
ρ3bJ1μ(s)=−3⟨ˉqq⟩256π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−3αβs)mcαβ2−4m2cms+(m2c(α+β)−3αβs)mqαβ],
ρ4aJ1μ(s)=⟨g2sGG⟩m2c4096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−2αβs)(1α3+1β3),
ρ4bJ1μ(s)=⟨g2sGG⟩m2c4096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)(m2c(α+β)−αβs)(3(m2c(α+β)−3αβs)αβ2−(3m2c(α+β)−5αβs)α2β),
ρ5aJ1μ(s)=3⟨ˉsgsσ⋅Gs⟩mc256π4∫αmaxαmindα∫βmaxβmindβ2m2c(α+β)−3αβsα,
ρ5bJ1μ(s)=3⟨ˉqgsσ⋅Gq⟩mc256π4∫αmaxαmindα∫βmaxβmindβ[(m2c(α+β)−2αβs)(1β−2(1−α−β)β2)+2mcmsβ],
ρ5cJ1μ(s)=⟨ˉsgsσ⋅Gs⟩768π4((s−m2c)ms−9m2cmq)√1−4m2cs+⟨ˉqgsσ⋅Gq⟩768π4((s−m2c)mq−9m2cms)√1−4m2cs,
ρ6aJ1μ(s)=⟨ˉss⟩⟨ˉqq⟩64π2(4m2c+2mcmq+mcms)√1−4m2cs,
Π6bJ1μ(M2B)=−⟨ˉss⟩⟨ˉqq⟩m3c32π2∫10dα(ms1−α+mqα)e−m2cα(1−α)M2B,
Π8J1μ(M2B)=m4c64π2∫10dα(⟨ˉqq⟩⟨ˉsgsσ⋅Gs⟩+⟨ˉss⟩⟨ˉqgsσ⋅Gq⟩(1−α)2M2B−2⟨ˉss⟩⟨ˉqgsσ⋅Gq⟩(1−α)m2c)e−m2cα(1−α)M2B,
4. Spectral densities for
J2μ :ρ0aJ2μ(s)=34096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2α3β3(m2c(α+β)−αβs)3(m2c(α+β)−5αβs),
ρ0bJ2μ(s)=−3mc1024π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−αβs)2((2m2c(α+β)−5αβs)mqα2β3+(m2c(α+β)−4αβs)msα3β2),
ρ3aJ2μ(s)=−3⟨ˉqq⟩256π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[4(1−α−β)(m2c(α+β)−2αβs)mcαβ2−4m2cms+(m2c(α+β)−3αβs)mqαβ],
ρ3bJ2μ(s)=−3⟨ˉss⟩256π4∫αmaxαmindα∫βmaxβmindβ(m2c(α+β)−αβs)[2(1−α−β)(m2c(α+β)−3αβs)mcα2β−4m2cmq+(m2c(α+β)−3αβs)msαβ],
ρ4aJ2μ(s)=⟨g2sGG⟩m2c4096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)2(m2c(α+β)−2αβs)(1α3+1β3),
ρ4bJ2μ(s)=⟨g2sGG⟩m2c4096π6∫αmaxαmindα∫βmaxβmindβ(1−α−β)(m2c(α+β)−αβs)(3(m2c(α+β)−3αβs)α2β−(3m2c(α+β)−5αβs)αβ2),
\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}}}\, ,
Exotic ˉD(∗)sD(∗) molecular states and scˉqˉc tetraquark states with JP=0+, 1+, 2+
- Received Date: 2021-05-07
- Available Online: 2021-09-15
Abstract: We have calculated the mass spectra for the