-
The history of multiquarks can be traced to 1964, when Gell-Mann and Zweig proposed such configurations in building the quark model [1, 2]. Although the existence of tetraquarks and pentaquarks has long been speculated, it has rarely, if ever, been proven. The scenario has changed since 2003, owing to the observation of numerous charmoniumlike/bottomoniumlike XYZ states [3], hidden-charm
Pc states [4–7], doubly-charmT+cc states [8, 9], and fully-charm tetraquark [10] states, which cannot be well explained within the traditional quark model. They are very good candidates for tetraquark and pentaquark states. Details regarding the experimental as well as theoretical progress can be found in review papers [11–17].In 2017, the LHCb Collaboration observed four
J/ψϕ structures, i.e.,X(4140) ,X(4274) ,X(4500) andX(4700) , in theB+→J/ψϕK+ decay process [18, 19], among whichX(4140) andX(4274) were confirmed to be consistent with previous measurements performed by the CDF Collaboration [20, 21], CMS Collaboration [22], D0 Collaboration [23], and BABAR Collaboration [24], whileX(4500) andX(4700) were new resonances. Inspired by these structures observed in theJ/ψϕ invariant mass spectrum,X(4140) andX(4274) were considered to be thecsˉcˉs tetraquark ground states, whereasX(4500) andX(4700) were interpreted as thecsˉcˉs tetraquark excited states, in various theoretical methods [25–33].Recently, the LHCb Collaboration performed an improved full amplitude analysis of the
B+→J/ψϕK+ decay using a signal yield 6 times larger than that previously analyzed [34]. They confirmed the fourJ/ψϕ states previously reported in Refs. [18, 19]. In addition, a newX(4685) state was observed in theJ/ψϕ final state with15σ significance, and its spin-parity was determined to beJP=1+ . Considering its observed channel, the quantum numbers ofX(4685) should beJPC=1++ with positive charge conjugation parity. Its mass and decay width are measured asm=4684±7+13−16 MeV andΓ=126±15+37−41 MeV. One may wonder ifX(4685) andX(4700) are the same resonance, as they were observed in the same final states with very similar masses and decay widths. However, LHCb determined their spin-parity asJP=1+ forX(4685) andJP=0+ forX(4700) [34]. They are definitely two different states.After the observations of the above
J/ψϕ resonances, there have been efforts to understand their underlying structures in the diquark-antidiquark picture. IfX(4140) andX(4274) can be assigned as the S-wavecsˉcˉs tetraquark ground states withJPC=1++ , theX(4685) state may be interpreted as the D-wavecsˉcˉs excited tetraquark state. In Ref. [28], the authors calculated the masses of the excited hidden-charm tetraquarks without internal diquark excitation (λ-mode excitation) by using the relativistic quark model. The mass of the D-wavecsˉcˉs tetraquark withJPC=1++ was calculated to be approximately 4.8 GeV. The same λ-mode excited1++ D-wavecsˉcˉs tetraquarks were also studied in the color flux-tube model with masses of approximately 4.9 and 5.2 GeV for theˉ3cs⊗3ˉcˉs and6cs⊗ˉ6ˉcˉs color structures, respectively [35]. In Ref. [29], the D-wavecsˉcˉs tetraquarks were investigated in different excitation modes by considering internal excited diquarks (ρ-mode excitation) in the relativistic quark model. The masses of the ρ-mode D-wavecsˉcˉs tetraquarks withJPC=1++ were predicted as 4.6–4.7 GeV, which are far lower than those of the λ-mode tetraquarks. Additionally, the authors of Ref. [36] calculated the mass of the ground state of the1++ S-wavecsˉcˉs tetraquark to be approximately 4.6 GeV according to the QCD sum rules, which is far higher than those obtained in Ref. [25]. In Ref. [37],X(4685) was also considered as the axialvector2S radial excitedcsˉcˉs tetraquark state.According to the above analyses and theoretical investigations, the newly observed
X(4685) state may be explained as a ρ-mode excitedD -wavecsˉcˉs tetraquark withJPC=1++ . In this work, we systematically study the mass spectra of the D-wavecsˉcˉs withJPC=1++ and1+− in both color symmetric6cs⊗ˉ6ˉcˉs and antisymmetricˉ3cs⊗3ˉcˉs configurations within the framework of QCD sum rules [38, 39]. We investigate the D-wave tetraquarks in different excitation structures, including the ρ-mode and λ-mode.The remainder of this paper is organized as follows. In Sec. II, we construct the nonlocal D-wave interpolating currents for
csˉcˉs tetraquark states withJPC=1++ and1+− in various excitation structures and color configurations. In Sec. III, we introduce the formalism of tetraquark QCD sum rules and calculate the two-point correlation functions and spectral densities for all currents. We perform numerical analyses to extract the full mass spectra of these D-wavecsˉcˉs tetraquark states in Sec. IV. The last section presents a summary. -
In this section, we construct the D-wave
csˉcˉs tetraquark interpolating currents withJPC=1++ and1+− . Thecsˉcˉs tetraquark is composed ofcs diquark andˉcˉs antidiquark fields. By analogy with the heavy baryon system, the orbital angular momentum of the tetraquark can be decomposed intoL=Lρ+Lλ=lρ1+lρ2+Lλ , wherelρ1 (lρ2 ) represents the internal orbital angular momentum for thecs (ˉcˉs ) field, andLλ represents the orbital angular momentum between the diquark and antidiquark fields. It is convenient to denote the orbital excitation of the tetraquark system as(Lλ,Lρ{lρ1,lρ2}) , as shown in Fig. 1. The D-wave excitedcsˉcˉs tetraquarks are the excitations withLρ+Lλ=2 . There exist several different excitation structures for the D-wave tetraquarks:(Lλ,Lρ{lρ1,lρ2})=(2,0{0,0}) ,(1,1{1,0}) ,(1,1{0,1}) ,(0,2{1,1}) ,(0,2{2,0}) ,(0,2{0,2}) . We study all these D-wave tetraquarks by constructing the interpolating currents with the same structures and quantum numbers.Figure 1. (color online) Excitation structure of the hidden-charm
csˉcˉs tetraquark system, in whichlρ1 (lρ2 ) represents the internal orbital angular momentum for thecs (ˉcˉs ) field, andLλ represents the orbital angular momentum between the diquark and antidiquark fields.The color structure of a diquark-antidiquark tetraquark operator
[cs][ˉcˉs] can be expressed viaSU(3) symmetry:(3⊗3)[cs]⊗(ˉ3⊗ˉ3)[ˉcˉs]=(6⊕ˉ3)[cs]⊗(3⊕ˉ6)[ˉcˉs]=(6⊗ˉ6)⊕(ˉ3⊗3)⊕(6⊗3)⊕(ˉ3⊗ˉ6)=(1⊕8⊕27)⊕(1⊕8)⊕(8⊕10)⊕(8⊕¯10),
(1) in which the color singlet structures come from the
6cs⊗ˉ6ˉcˉs andˉ3cs⊗3ˉcˉs terms, which are denoted as the color symmetric and antisymmetric configurations, respectively. In this work, we consider both these color configurations. We use only the S-wave good diquark fieldOS=cTaCγ5sb withJP=0+ to compose the D-wavecsˉcˉs tetraquark currents by inserting covariant derivative operators. For example, one can obtain a ρ-mode P-wave diquark field withJP=1− OP,μ=cTaCγ5Dμsb,
(2) and a ρ-mode D-wave diquark field with
JP=2+ OD,μν=cTaCγ5DμDνsb,
(3) where
Dμ=∂μ+igsAμ is the covariant derivative, the subscriptsa,b are color indices,C denotes the charge conjugate operator, and T represents the transpose of the quark fields. The corresponding charge conjugate antidiquark fields areˉOS=ˉcaCγ5ˉsTb,ˉOP,μ=ˉcaCγ5DμˉsTb,ˉOD,μν=ˉcaCγ5DμDνˉsTb.
(4) To compose the λ-mode excited tetraquark operator, one should insert the covariant derivative operator between the diquark and antidiquark fields.
Lλ=0:OSˉOS,Lλ=1:OSDμˉOS,Lλ=2:OSDμDνˉOS.
(5) Considering both the symmetric and antisymmetric color configurations, we construct the D-wave
csˉcˉs interpolating tetraquark currents withJPC=1++ asJA1,μν=[cTaCγ5sb]{Dμ,Dν}([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])+[ˉcaCγ5ˉsTb]{Dμ,Dν}([cTaCγ5sb]−[cTbCγ5sa]),JS1,μν=[cTaCγ5sb]{Dμ,Dν}([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])+[ˉcaCγ5ˉsTb]{Dμ,Dν}([cTaCγ5sb]+[cTbCγ5sa]),JA2,μν=[cTaCγ5Dμsb]Dν([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])+[ˉcaCγ5DμˉsTb]Dν([cTaCγ5sb]−[cTbCγ5sa]),JS2,μν=[cTaCγ5Dμsb]Dν([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])+[ˉcaCγ5DμˉsTb]Dν([cTaCγ5sb]+[cTbCγ5sa]),JA3,μν=[cTaCγ5sb]Dμ([ˉcaCγ5DνˉsTb]−[ˉcbCγ5DνˉsTa])+[ˉcaCγ5ˉsTb]Dμ([cTaCγ5Dνsb]−[cTbCγ5Dνsa]),JS3,μν=[cTaCγ5sb]Dμ([ˉcaCγ5DνˉsTb]+[ˉcbCγ5DνˉsTa])+[ˉcaCγ5ˉsTb]Dμ([cTaCγ5Dνsb]+[cTbCγ5Dνsa]),JA4,μν=[cTaCDμγ5sb]([ˉcaCγ5DνˉsTb]−[ˉcbCγ5DνˉsTa])+[ˉcaCγ5DμˉsTb]([cTaCγ5Dνsb]−[cTbCγ5Dνsa]),JS4,μν=[cTaCDμγ5sb]([ˉcaCγ5DνˉsTb]+[ˉcbCγ5DνˉsTa])+[ˉcaCγ5DμˉsTb]([cTaCγ5Dνsb]+[cTbCγ5Dνsa]),JA5,μν=[cTaCγ5DμDνsb]([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])+[ˉcaCγ5DμDνˉsTb]([cTaCγ5sb]−[cTbCγ5sa]),JS5,μν=[cTaCγ5DμDνsb]([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])+[ˉcaCγ5DμDνˉsTb]([cTaCγ5sb]+[cTbCγ5sa]),JA6,μν=[cTaCγ5sb]([ˉcaCγ5DμDνˉsTb]−[ˉcbCγ5DμDνˉsTa])+[ˉcaCγ5ˉsTb]([cTaCγ5DμDνsb]−[cTbCγ5DμDνsa]),JS6,μν=[cTaCγ5sb]([ˉcaCγ5DμDνˉsTb]+[ˉcbCγ5DμDνˉsTa])+[ˉcaCγ5ˉsTb]([cTaCγ5DμDνsb]+[cTbCγ5DμDνsa]), (6) and the D-wave
csˉcˉs interpolating tetraquark currents withJPC=1+− asJA7,μν=[cTaCγ5sb]{Dμ,Dν}([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])−[ˉcaCγ5ˉsTb]{Dμ,Dν}([cTaCγ5sb]−[cTbCγ5sa]),JS7,μν=[cTaCγ5sb]{Dμ,Dν}([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])−[ˉcaCγ5ˉsTb]{Dμ,Dν}([cTaCγ5sb]+[cTbCγ5sa]),JA8,μν=[cTaCγ5Dμsb]Dν([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])−[ˉcaCγ5DμˉsTb]Dν([cTaCγ5sb]−[cTbCγ5sa]),JS8,μν=[cTaCγ5Dμsb]Dν([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])−[ˉcaCγ5DμˉsTb]Dν([cTaCγ5sb]+[cTbCγ5sa]),JA9,μν=[cTaCγ5sb]Dμ([ˉcaCγ5DνˉsTb]−[ˉcbCγ5DνˉsTa])−[ˉcaCγ5ˉsTb]Dμ([cTaCγ5Dνsb]−[cTbCγ5Dνsa]),JS9,μν=[cTaCγ5sb]Dμ([ˉcaCγ5DνˉsTb]+[ˉcbCγ5DνˉsTa])−[ˉcaCγ5ˉsTb]Dμ([cTaCγ5Dνsb]+[cTbCγ5Dνsa]),JA10,μν=[cTaCDμγ5sb]([ˉcaCγ5DνˉsTb]−[ˉcbCγ5DνˉsTa])−[ˉcaCγ5DμˉsTb]([cTaCγ5Dνsb]−[cTbCγ5Dνsa]),JS10,μν=[cTaCDμγ5sb]([ˉcaCγ5DνˉsTb]+[ˉcbCγ5DνˉsTa])−[ˉcaCγ5DμˉsTb]([cTaCγ5Dνsb]+[cTbCγ5Dνsa]),JA11,μν=[cTaCγ5DμDνsb]([ˉcaCγ5ˉsTb]−[ˉcbCγ5ˉsTa])−[ˉcaCγ5DμDνˉsTb]([cTaCγ5sb]−[cTbCγ5sa]),JS11,μν=[cTaCγ5DμDνsb]([ˉcaCγ5ˉsTb]+[ˉcbCγ5ˉsTa])−[ˉcaCγ5DμDνˉsTb]([cTaCγ5sb]+[cTbCγ5sa]),JA12,μν=[cTaCγ5sb]([ˉcaCγ5DμDνˉsTb]−[ˉcbCγ5DμDνˉsTa])−[ˉcaCγ5ˉsTb]([cTaCγ5DμDνsb]−[cTbCγ5DμDνsa]),JS12,μν=[cTaCγ5sb]([ˉcaCγ5DμDνˉsTb]+[ˉcbCγ5DμDνˉsTa])−[ˉcaCγ5ˉsTb]([cTaCγ5DμDνsb]+[cTbCγ5DμDνsa]),
(7) where
{Dμ,Dν}=DμDν+DνDμ . The interpolating currents with the superscripts "S" and "A" denote the symmetric[cs]6[ˉcˉs]ˉ6 and antisymmetric[cs]ˉ3[ˉcˉs]3 color structures, which are abbreviated as3 and6 , respectively, hereinafter. The excitation structures(Lλ,Lρ{lρ1,lρ2}) , color configurations, andJPC quantum numbers for these interpolating currents are presented in Table 1. The abbreviation3λλ/6λλ (3ρρ/6ρρ ) indicates that the corresponding current contains two λ-orbital (ρ-orbital) momentums with an antisymmetric/symmetric color structure, while3λρ/6λρ indicates that the current contains one λ-orbital momentum and one ρ-orbital momentum with an antisymmetric/symmetric color structure. In the following, we investigate the mass spectra for the D-wavecsˉcˉs tetraquarks by using these interpolating currents. Among the currents belonging to the(0,2{2,0}) and(0,2{0,2}) structures, we only study the(0,2{2,0}) ones, because the(0,2{0,2}) currents would yield the same results in our calculations.(Lλ,Lρ{lρ1,lρ2}) [cs]ˉ3[ˉcˉs]3 [cs]6[ˉcˉs]ˉ6 JPC (2,0{0,0}) JA1,μν(3λλ) JS1,μν(6λλ) 1++ JA7,μν(3λλ) JS7,μν(6λλ) 1+− (1,1{1,0}) JA2,μν(3λρ) JS2,μν(6λρ) 1++ JA8,μν(3λρ) JS8,μν(6λρ) 1+− (1,1{0,1}) JA3,μν(3λρ) JS3,μν(6λρ) 1++ JA9,μν(3λρ) JS9,μν(6λρ) 1+− (0,2{1,1}) JA4,μν(3ρρ) JS4,μν(6ρρ) 1++ JA10,μν(3ρρ) JS10,μν(6ρρ) 1+− (0,2{2,0}) JA5,μν(3ρρ) JS5,μν(6ρρ) 1++ JA11,μν(3ρρ) JS11,μν(6ρρ) 1+− (0,2{0,2}) JA6,μν(3ρρ) JS6,μν(6ρρ) 1++ JA12,μν(3ρρ) JS12,μν(6ρρ) 1+− Table 1. Excitation structures, color configurations, and
JPC quantum numbers for the D-wavecsˉcˉs interpolating currents given by Eqs. (6) and (7). -
In this section, we introduce the method of QCD sum rules for the hidden-charm tetraquark states. The two-point correlation functions for the tensor currents can be written as
Πμν,ρσ(q2)=i∫d4xeiq⋅x⟨0|T[Jμν(x)J†ρσ(0)]|0⟩=T+μνρσΠ1(q2)+⋯,
(8) where
T±μνρσ=(qμqνq2ηνσ±(μ↔ν))±(ρ↔σ),ημν=qμqνq2−gμν,
(9) Π1(q2) is the polarization function related to the spin-1 intermediate state, and"⋯" represents other tensor structures relating to different hadron states. The tensor current can couple to the spin-1 physical state X through⟨0|Jμν(x)|1PC(→p,r)⟩=Zϵμναβ∈α(→p,r)pβ,⟨0|Jμν(x)|1(−P)C(→p,r)⟩=Z+(∈μ(→p,r)pν+∈ν(→p,r)pμ)+Z−(∈μ(→p,r)pν−∈ν(→p,r)pμ),
(10) where
Z,Z+,Z− are coupling constants,ϵμναβ is the antisymmetical tensor, and∈α is the polarization tensor.At the hadron level, the two-point correlation function can be written as
Π(q2)=1π∫∞s<ImΠ(s)s−q2−iϵds,
(11) where we use the form of the dispersion relation, and
s< denotes the physical threshold. The imaginary part of the correlation function is defined as the spectral function, which is usually evaluated at the hadron level by inserting intermediate hadron states∑n|n⟩⟨n| ρ(s)≡1πImΠ(s)=∑nδ(s−M2n)⟨0|η|n⟩⟨n|η†|0⟩=f2Xδ(s−m2X)+continuum,
(12) where we have adopted the usual parametrization of one-pole dominance for the ground state X and a continuum contribution. Researchers have investigated the excited mesons [40–42], baryons [43], and tetraquarks [44–46] in QCD sum rules by using the non-local interpolating currents under the "pole+continuum" approximation. The spectral density
ρ(s) can also be evaluated at the quark-gluon level via the operator product expansion (OPE). To pick out the contribution of the lowest lying resonance in (12), the QCD sum rules are established asLk(s0,M2B)=f2Xm2kHe−m2H/M2B=∫s04m2cdse−s/M2Bρ(s)sk,
(13) where
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 asmX(s0,M2B)=√L1(s0,M2B)L0(s0,M2B),
(14) which is the function of two parameters
M2B ands0 . We discuss the details of obtaining suitable parameter working regions in QCD sum rule analyses in next section. Using the operator production expansion method, the two-point function can also be evaluated at the quark-gluonic level as a function of various QCD parameters, such as QCD condensates, quark masses, and the strong coupling constantαs . To evaluate the Wilson coefficients, we adopt the heavy quark propagator in the momentum space and the strange quark propagator in the coordinate space:iSabc(p)=iδabˆp−mc+i4gsλnab2Gnμνσμν(ˆp+mc)+(ˆp+mc)σμν12+iδab12⟨g2sGG⟩mcp2+mcˆp(p2−m2c)4,iSabs(x)=iδab2π2x4ˆx−δab12⟨ˉss⟩+i32π2λnab2gsGnμν1x2(σμνˆx+ˆxσμν)+δabx2192⟨ˉsgsσ⋅Gs⟩−msδab4π2x2+iδabms⟨ˉss⟩48ˆx−ims⟨ˉsgsσ⋅Gs⟩δabx2ˆx1152,
(15) where
ˆp=pμγμ andˆx=xμγμ . In this work, we evaluate the Wilson coefficients of the correlation function up to dimension ten condensates at the leading order ofαs . We find that the calculations are highly complex owing to the existence of the covariant derivative operators. The results of spectral functions are too lengthy to present here; thus, they are provided in the Appendix. -
In this section, we perform the QCD sum rule analyses for the
csˉcˉs tetraquark systems. We use the following values of the quark masses and various QCD condensates [3, 47–55]:mc(mc)=1.27±0.02GeV,mc/ms=11.76+0.05−0.10,⟨ˉqq⟩=−(0.24±0.03)3GeV3,⟨ˉqgsσ⋅Gq⟩=−M20⟨ˉqq⟩,⟨ˉqqˉqq⟩=⟨ˉqq⟩2,M20=(0.8±0.2)GeV2,⟨ˉss⟩/⟨ˉqq⟩=0.8±0.1,⟨g2sGG⟩=(0.48±0.14)GeV4,
(16) where the charm quark mass
mc is the "running" mass in the¯MS scheme. To ensure the unified renormalization scale in our analyses, we use the renormalization scheme and scale independentmc/ms mass ratio from PDG [3] to obtain the strange quark massms .To establish a stable mass sum rule, one should initially find the appropriate parameter working regions, i.e, for the continuum threshold
s0 and the Borel massM2B . The thresholds0 can be determined via the minimized variation of the hadronic massmX with respect to 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 is defined in Eq. (13).As an example, we use the color antisymmetric current
JA5,μν(x) withJPC=1++ in the(0,2{2,0}) excitation mode 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, which is proportional to the charm quark massmc . Figure 2 shows the contributions of the perturbative term and various condensate terms to the correlation function with respect toM2B whens0 tends to infinity. It is clear that the Borel massM2B should be large enough to ensure the convergence of the OPE series. In this work, we require that the perturbative term be two times larger than the quark condensate term, providing the lower bound of the Borel massM2B≥2.82GeV2 . The other QCD condensates are far smaller than the quark condensate in this region ofM2B . Studying the pole contribution defined in Eq. (17) reveals that the PC is very small for such D-wavecsˉcˉs tetraquark systems owing to the high dimension of the interpolating current. To find an upper bound on the Borel mass, we require the pole contribution to be larger than20% . As a result, the reasonable Borel window for the currentJA5,μν(x) is obtained as2.94GeV2≤M2B≤3.90GeV2 .Figure 2. (color online) Contributions of various OPE terms to the correlation function for the current
JA5,μν(x) as a function ofM2B whens0→∞ .As mentioned previously, the variation of the extracted hadron mass
mX with respect toM2B should be minimized to obtain the optimal value of the continuum thresholds0 . We show the variation ofmX withs0 in the left panel of Fig. 3, from which the optimized value of the continuum threshold can be chosen ass0≈(30.0±1.5)GeV2 . In the right panel of Fig. 3, the mass sum rules are established to be very stable in the above parameter regions ofs0 andM2B . The hadron mass for this D-wavecsˉcˉs tetraquark withJPC=1++ can be obtained asmJA5=5.16+0.12−0.13GeV,
(18) where the errors come from the uncertainties of the threshold
s0 , Borel massM2B , quark masses, and various QCD condensates in Eq. (16). Performing the same numerical analyses for all the interpolating currents in Eqs. (6)–(7), we find that only the currentsJS5,μν(x) ,JA(S)11,μν(x) , andJA(S)4,μν(x) withJPC=1++ exhibit the same mass sum rule behaviors asJA5,μν(x) . We present the numerical results in Table 2.(Lλ,Lρ{lρ1,lρ2}) Current JPC mA/GeV s0,A/GeV2 M2B,A/GeV2 PCA(%) mS/GeV s0,S/GeV2 M2B,S/GeV2 PCS(%) (2,0{0,0}) JA(S)1,μν 1++ 4.70+0.12−0.11 27(±5%) 3.27∼3.92 27.3 4.91+0.11−0.12 28(±5%) 3.56∼4.20 26.5 (2,0{0,0}) JA(S)7,μν 1+− 4.78+0.12−0.11 27(±5%) 3.58∼4.16 25.4 4.89+0.10−0.11 28(±5%) 3.60∼4.50 28.5 (1,1{1,0}) JA(S)2,μν 1++ 4.80+0.12−0.16 28(±5%) 3.15∼3.94 39.6 4.84+0.12−0.16 29(±5%) 2.63∼4.13 37.9 (1,1{1,0}) JA(S)8,μν 1+− 4.81±0.10 27(±5%) 3.71∼4.51 26.3 4.85+0.11−0.10 28(±5%) 4.69∼5.16 28.2 (1,1{0,1}) JA(S)3,μν 1++ 4.80+0.11−0.10 26(±5%) 2.75∼3.31 26.1 4.82+0.12−0.11 27(±5%) 3.37∼4.11 47.0 (1,1{0,1}) JA(S)9,μν 1+− 4.98+0.13−0.23 26(±5%) 2.73∼3.14 24.0 4.92+0.11−0.10 28(±5%) 3.55∼3.91 23.4 (0,2{1,1}) JA(S)4,μν 1++ 4.80+0.10−0.11 26(±5%) 2.51∼3.14 27.5 4.80+0.10−0.11 26(±5%) 2.52∼3.15 27.4 (0,2{1,1}) JA(S)10,μν 1+− 4.83+0.10−0.11 28(±5%) 3.06∼3.82 28.6 4.83+0.10−0.12 28(±5%) 3.08∼3.82 28.3 (0,2{2,0}) JA(S)5,μν 1++ 5.16+0.12−0.13 30(±5%) 2.94∼3.90 41.4 4.69±0.09 24(±5%) 2.22∼2.82 27.5 (0,2{2,0}) JA(S)11,μν 1+− 5.19+0.12−0.13 30(±5%) 3.55∼3.92 43.4 4.67±0.09 23(±5%) 2.69∼2.87 21.6 (1,1)mix JA(S)2,μν+JA(S)3,μν 1++ 4.80±0.10 27(±5%) 3.01∼3.76 24.1 4.93+0.09−0.10 29(±5%) 3.22∼4.02 38.4 (1,1)mix JA(S)8,μν+JA(S)9,μν 1+− 4.80+0.11−0.13 26(±5%) 2.71∼3.13 30.2 4.94±0.10 29(±5%) 3.37∼4.21 38.2 Table 2. Hadron masses of the
csˉcˉs tetraquark states with differentJPC quantum numbers and(Lλ,Lρ{lρ1,lρ2}) excitation structures. The subscripts "A" and "S" denote the numerical results for the color antisymmetric and symmetric currents, respectively.Except for
JA(S)5,μν(x) ,JA(S)11,μν(x) , andJA(S)4,μν(x) , the interpolating currents exhibit very different mass sum rule behaviors. As shown in the left panel forJS1,μν(x) , the extracted hadron mass increases monotonically with the continuum thresholds0 . Thus, one is not able to find an optimal value ofs0 to minimize the variation of the hadron mass with respect toM2B . For such a situation, we define the following hadron massˉmX and quantityχ2(s0) to study the stability of the mass sum rules:ˉmX(s0)=N∑i=1mX(s0,M2B,i)N,
(19) χ2(s0)=N∑i=1[mX(s0,M2B,i)ˉmX(s0)−1]2,
(20) where
M2B,i(i=1,2,…,N) represents N definite values for the Borel parameterM2B in the Borel window. According to the above definition, the optimal choice for the continuum thresholds0 in the QCD sum rule analysis can be obtained by minimizing the quantityχ2(s0) , which is a function of onlys0 . This relation is shown in the right panel of Fig. 4, in which there is a minimum point at approximatelys0≈28.0GeV2 . We can thus determine the working range for the continuum threshold to bes0=(28.0±1.4)GeV2 , as shown in the left panel of Fig. 4. The hadron mass is thus obtained asFigure 4. (color online) Mass curves (left) and
χ2 curve (right) for the currentJS1,μν(x) withJPC=1++ .mJS1=4.91+0.11−0.12GeV.
(21) In these analyses, we find that the OPE series for the
JA(S)4,μν(x) andJA(S)10,μν(x) belonging to the(0,2{1,1}) structure differ significantly from those of other interpolating currents. As shown in the Appendix, the quark condensate does not contribute to the correlation function for any of the(0,2{1,1}) currents.By performing similar analyses, we obtain the numerical results for all the other interpolating currents in Eqs. (6) and (7), and they are presented in Table 2. The extracted hadron masses from
JA1,μν(x) andJS5,μν(x) withJPC=1++ agree well with the mass of the newly observed resonanceX(4685) , implying thatX(4685) can be interpreted as a D-wavecsˉcˉs tetraquark state withJPC=1++ in the excitation mode of(2,0{0,0}) or(0,2{2,0}) .JPC S-wave P-wave 1++ D∗s0ˉDs1,DsˉD∗s,DsˉD∗s1, DsˉDs1,D∗s0ˉD∗s,D∗s0ˉD∗s1, Ds1ˉDs1,Ds1ˉD∗s2, D∗sˉDs1,D∗s1ˉDs1,D∗sˉD∗s2, J/ψϕ D∗s1ˉD∗s2,hc(1P)ϕ 1+− D∗s0ˉDs1,DsˉD∗s,DsˉD∗s1, DsˉDs1,D∗s0ˉD∗s,D∗s0ˉD∗s1, Ds1ˉDs1,Ds1ˉD∗s2, D∗sˉDs1,D∗s1ˉDs1,D∗sˉD∗s2, ηcϕ χc0(1P)ϕ,χc1(1P)ϕ Table 3. Possible decay channels of the D-wave
csˉcˉs tetraquark states withJPC=1++ and1+− .Considering the same physical picture for the
(1,1{1,0}) and(1,1{0,1}) excitation structures, the interpolating currentsJA(S)2,μν(x) andJA(S)3,μν(x) exhibit similar mass sum rules. The currentsJA2,μν(x) andJA3,μν(x) give almost degenerate hadron masses, as shown in Table 2. To study their mixing effects, we also perform analyses for the mixed currentsJA(S)2,μν+JA(S)3,μν . Our calculations show that the off-diagonal correlatorΠA(S)23(q2) is nonzero, implying that the currentsJA2,μν(x) andJA3,μν(x) may couple to the same hadron state. The same situation arises for the interpolating currentsJA(S)8,μν(x) andJA(S)9,μν(x) , which couple to the same tetraquark state. -
We investigated the mass spectra for the D-wave
csˉcˉs tetraquark states withJPC=1++ and1+− in the framework of QCD sum rules. We constructed the D-wave non-local interpolating tetraquark currents with covariant derivative operators in the (Lλ,Lρ{lρ1,lρ2})=(2,0{0,0}),(1,1{1,0}),(1,1{0,1}),(0,2{1,1}),(0,2{2,0}),(0,2{0,2}) excitation structures. The two-point correlation functions were calculated up to dimension ten condensates in the leading order ofαs . We established reliable mass sum rules for all these currents and obtained the mass spectra of D-wavecsˉcˉs tetraquarks, as shown in Table 2. Our results support the interpretation of the recently observedX(4685) structure as a D-wavecsˉcˉs tetraquark state withJPC=1++ in the(2,0{0,0}) or(0,2{2,0}) excitation mode. However, some other possibilities of the excitation modes cannot be excluded by our results within errors.The mass spectra of
csˉcˉs tetraquark states in different color configurations were studied in Ref. [35], and the results indicated that the masses of color symmetric tetraquarks are lower than those of color antisymmetric tetraquarks in the ground state (L=0 ). Similar results were obtained for the fully heavy tetraquark states [56–58]. However, the situation is different for the excitedcsˉcˉs tetraquarks: the masses of color antisymmetric tetraquarks are lower than those of color symmetric tetraquarks. Such behavior is consistent with our results in Table 2 for the D-wavecsˉcˉs tetraquarks, except for those in the(0,2{2,0}) structures with two ρ-mode excitations. In Table 2, the masses for the positiveC -parity tetraquarks follow the relation6ρρ<3λλ<3λρ<3ρρ , and those for the negativeC -parity tetraquarks exhibit the relation6ρρ<3λλ<6λλ<3ρρ , which is consistent with the conclusion for P-waveccˉcˉc systems [57].We present the mass spectra of these
csˉcˉs tetraquarks in comparison with the corresponding two-meson open-charm mass thresholds in Fig. 5. Clearly, these D-wavecsˉcˉs tetraquarks withJPC=1++ and1+− lie above the mass thresholds ofDsˉD∗s ,J/ψϕ , andηcϕ . Accordingly, we present their possible decay channels in both the S-wave and P-wave in Table 3. We suggest searching for these D-wavecsˉcˉs tetraquarks in both the hidden-charm channelsJ/ψϕ andηcϕ , as well as open-charm channels such asDsˉD∗s andDsˉD∗s1 . -
The spectral functions for the D-wave interpolating current
JA(S)i can be written asρi;A(S)(s)=ρperti;A(S)(s)+⟨ˉss⟩ρ⟨ˉss⟩i;A(S)(s)+ms⟨ˉss⟩ρms⟨ˉss⟩i;A(S)(s)+⟨g2sG2⟩ρ⟨g2sG2⟩i;A(S)(s)+⟨ˉsσ⋅Gs⟩ρ⟨ˉsσ⋅Gs⟩i;A(S)(s)+ms⟨ˉsσ⋅Gs⟩ρms⟨ˉsσ⋅Gs⟩i;A(S)(s)+⟨ˉssˉss⟩ρ⟨ˉssˉss⟩i;A(S)(s)+⟨ˉss⟩⟨ˉsσ⋅Gs⟩ρ⟨ˉss⟩⟨ˉsσ⋅Gs⟩i;A(S)(s)+⟨g2sG2⟩⟨ˉss⟩ρ⟨g2sG2⟩⟨ˉss⟩i;A(S)(s)+ms⟨g2sG2⟩⟨ˉss⟩ρms⟨g2sG2⟩⟨ˉss⟩i;A(S)(s)+⟨g2sG2⟩2ρ⟨g2sG2⟩2i;A(S)(s)+⟨g2sG2⟩⟨ˉsσ⋅Gs⟩ρ⟨g2sG2⟩⟨ˉsσ⋅Gs⟩i;A(S)(s)+ms⟨g2sG2⟩⟨ˉsσ⋅Gs⟩ρms⟨g2sG2⟩⟨ˉsσ⋅Gs⟩i;A(S)(s). (22) The spectral functions for the
(2,0{0,0}) structure are given as follows:ρpert1,7;A(S)(s)=−∫xmax0dx∫ymaxymindyx1612800π5(y−1)5F(s,x,y)3c1(2(x−1)(10((x(5(13x−42)x+273)−140)x+35)y2−28(39x2−45x+20)y−21x(2x+5)+(10((x(2(x−35)x+189)−140)x+35)y2+28(x((15x−74)x+70)−20)y+21((23x−30)x+10))cp+210)x2yF(s,x,y)3+42x((x−1)(y−1)(10((x(5(13x−42)x+273)−140)x+35)y4−((x(50x+923)−1165)x+590)y3+2((32x−165)x+190)y2+60(x)−2)y+10sx+(−4((x((59x−184)x+195)−90)x+15)xy3+((x(58(x−3)x+195)−120)x+30)y2+2(3(16x−45)x+110)xy−60y+5(3x−8)x+30)mcms+cp((x−1)(y−1)(10((x(2(x−35)x+189)−140)x+35)y2+((450x−2183)x+2065)xy−590y+24(23x−30)x+240)sxy2+(−4(x(4(x−5)x+35)−15)(x−1)xy3+(((4(64−13x)x−245)x+20)x+30)y2−2(37x3−60x+30)y+55x2−80x+30)mcms))F(s,x,y)2+15(y−1)((x−1)(y−1)(50((x(5(13x−42)x+273)−140)x+35)y4−14((x(10x+361)−455)x+230)y3+7((79x−310)x+330)y2+420(x−2)y+70)s2yx2+14mcms((−11((x((59x−184)x+195)−90)x+15)xy4+((x(2(96x−325)x+765)−440)x+105)y3+(x((153x−485)x+430)−150)y2+5((7x−12)x+12)y+5(x−3))sx+(6(4(2x((2x−5)x+5)−5)x+5)y2
+5(7(x−4)x+30)xy−60y+10(x−3)x+30)mcms)+cp((x−1)(y−1)(50((x(2(x−35)x+189)−140)x+35)y2+14(x((180x−851)x+805)−230)y+147((23x−30)x+10))s2x2y3+7mcms((−22(x(4(x−5)x+35)−15)(x−1)xy3−2(x(((143x−680)x+595)x+20)−105)y2−((x(359x+240)−930)x+420)y+35((11x−16)x+6))sxy+2(2(4(2x((x−5)x+10)−15)x+15)y2+5(11(x−4)x+42)xy−60y+10(7x−9)x+30)mcms)))F(s,x,y)+60(y−1)2s((x−1)(y−1)(4((x(5(13x−42)x+273)−140)x+35)y3−56(2(4x−5)x+5)y2+21((3x−10)x+10)y+35(x−2))s2x2y3+2((x−1)(y−1)(2((x(2(x−35)x+189)−140)x+35)y2+14(x((8x−37)x+35)−10)y+7(23x−30)x+70)s2x2y3+7(2(2(4(2x((x−5)x+10)−15)x+15)y2+5(11(x−4)x+42)xy−60y+10(7x−9)x+30)mcms−sxy(xy−1)(55x2−80x+2(x(4(x−5)x+35)−15)(x−1)y2+2(x((17x−80)x+90)−30)y+30))mcms)cpy+7mcms((−4((x((59x−184)x+195)−90)x+15)xy4+4((x−2)((17x−25)x+25)x+15)y3+(x((53x−210)x+250)−120)y2+5((7x−20)x+18)y+20x−30)sxy+2(12(4(2x((2x−5)x+5)−5)x+5)y3+10(x(5(x−4)x+24)−12)y2+30((x−3)x+3)y+15(x−2))mcms))),ρ⟨ˉss⟩1,7;A(S)(s)=−∫xmax0dx∫ymaxymindyc1xmc96π3(y−1)4F(s,x,y)2(2s(y−1)((x−1)y2((x(26x−1)−14)cp−22x2+50x−10)+11xy4(x3(cp−23)−2x(cp+5)+cp+7x4+24x2+1)−y3((x(x(11x+37)−19)−7)(x−1)cp−2((21x−40)x+28)x+7)−(x−1)y(7(x−1)cp+10x−4)−x+1)F(s,x,y)+((x−1)(y−1)cp(2(x2+x−1)xy2+x(2−5x)y+x+y−1)+2((x((7x−23)x+24)−10)x+1)xy3+(x((5x−9)x+7)−1)y2−(x−1)((5x−9)x+2)y−(x−1)2)F(s,x,y)2+6s2(y−1)2y(−2y3((x(x+2)−4)x2cp+cp−2(2(x−2)x+3)x+1)+(x−1)y2(2(2x2+x−2)cp+x(11−4x)−4)+2xy4(x3(cp−23)−2x(cp+5)+cp+7x4+24x2+1)−(x−1)2y(2cp+3)−x+1)),ρ⟨msˉss⟩1,7;A(S)(s)=−∫xmax0dx∫ymaxymindyc196π3(y−1)3F(s,x,y)(−2F(s,x,y)2(m2c(5x2cp+y2(x4(cp+9)−4x3(cp+5)+2x2(5cp+9)−8x(cp+1)+2(cp+1))+2y((x((x−7)x+7)−2)cp+((x−5)x+5)x−2)−6xcp+2cp+x2−2x+2)−s(x−1)x(y−1)(35y4(((x((x−4)x+10)−8)x+2)cp+(3x((x−4)x+6)−8)x+2)+y3(59(x((x−7)x+7)−2)cp+(233−x(81x+233))x−118)+2y2(12((5x−6)x+2)cp+(14x−33)x+38)+12(x−2)y+2))−3s(y−1)F(s,x,y)(2m2c(2y3(x4(cp+9)−4x3(cp+5)+2x2(5cp+9)−8x(cp+1)+2(cp+1))+4y2((x((x−7)x+7)−2)cp+x(x−2)2−2)+xy(2(5x−6)cp+3(x−2))+y(4cp+6)+x−2)−s(x−1)x(y−1)y(25y4(((x((x−4)x+10)−8)x+2)cp+(3x((x−4)x+6)−8)x+2)+2y3(23(x((x−7)x+7)−2)cp+(91−x(27x+91))x−46)+y2(21((5x−6)x+2)cp+(29x−62)x+66)+12(x−2)y+2))+(x−1)xy(5y2(((x((x−4)x+10)−8)x+2)cp+(3x((x−4)x+6)−8)x+2)+4y(2(x((x−7)x+7)−2)cp−3(x(x+3)−3)x−4)+3((5x−6)x+2)cp−3x+6)F(s,x,y)3+6s3(x−1)x(y−1)3y3(2y3(((x((x−4)x+10)−8)x+2)cp+(3x((x−4)x+6)−8)x+2)+4y2((x((x−7)x+7)−2)cp−(x(x+4)−4)x−2)+xy(2(5x−6)cp+3(x−2))+y(4cp+6)+x−2)),ρ⟨gsG2⟩1,7;A(S)(s)=∫xmax0dx∫ymaxymindyx2c1mc1935360π5(x−1)3(y−1)5(−2(x−1)(2(y((x((x−3)x+3)y−3)y+3)−1)(10((2cp+65)x4−70(cp+3)x3+21(9cp+13)x2−140(cp+1)x+35(cp+1))y2+28(−39x2+45x+(x((15x−74)x+70)−20)cp−20)y+21((2x+5)(−x)+((23x−30)x+10)cp+10))xymc+21(−4
\begin{aligned}[b] & \left.\Big(\left(\left(x \left(4 x^2\right.\right.\right.\right.-34 x+105\Big)-125\Big) x+45\Big) c_p (x-1)^2+(x ((x ((59 x-327) x+762)-949) x+660)-255) x\\ & +45\Big) x y^5+\left(3 (x (2 ((-7 (x-8) x-116) x+61) x+65)-120) x+(150-x ((4 (x (23 (x-8) x+494)-554) x\right.\\ & \left.+805) x+220)) c_p+150\right) y^4-2 \left((3 x ((-5 (x-13) x-218) x+300)-650) x+((x ((5 (17 x-59) x+141) x\right.\\ & \left.+535)-660) x+210) c_p+210\right) y^3+2 \left((x ((50 x-279) x+570)-540) x+((x (4 (5 x-61) x+725)-690)\right.\\ & x+210)c_p+210\Big) y^2+6 \left(((11 x-45) x+60) x+2 (x (8 (x-5) x+45)-15) c_p-30\right) y+5 \left((3 x-8) x\right.\\ & \left.\left.\left.\left.+((11 x-16) x+6) c_p+6\right)\right) m_s\right) F\left(s,x,y\right){}^3+21 \left(2 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(4 \left((x ((59 x-184)\right.\right.\right.\right.\\ & \left.x+195)-90) x+(x (4 (x-5) x+35)-15) (x-1) c_p+15\right) x y^3+\left(((-58 (x-3) x-195) x+120) x+(x\right.\\ & \left.(x (4 (13 x-64) x+245)-20)-30) c_p-30\right) y^2+2 \left((3 (45-16 x) x-110) x+\left(37 x^3-60 x+30\right) c_p+30\right)\\& \left.\left.\left.y-30 \left(c_p+1\right)-5 x^2 \left(11 c_p\right.+3\Big)+40 \left(2 c_p+1\right) x\right) m_s m_c^2-2 s (x-1) x (y-1) (y ((x ((x-3) x+3) y-3) y\right.\right. \\ & \left.+3)-1) \left(10 \left(\left(2 c_p+65\right) x^4-70\right.\right.\left(c_p+3\right) x^3+21 \left(9 c_p+13\right) x^2-140 \left(c_p+1\right) x+35 \left(c_p+1\right)\right) y^4\\ & \left.\left.\left.+\left(50 \left(9 c_p-1\right) x^3-\left(2183 c_p+923\right) x^2\right.+5 \left(413 c_p+233\right) x-590 \left(c_p+1\right)\right) y^3+\left(8 \left(69 c_p+8\right) x^2\right.\right.\right.\\& \left.\left.\left.\left.-30 \left(24 c_p+11\right) x+240 c_p+380\right) y^2+60 (x-2) y+10\right) m_c+3 (x-1) (y-1) \left(22 \left(4 c_p+59\right) y^6 x^7\right.\right.\right.\\ & \left.\left.\left.\left.+2 \left(-33 \left(14 c_p+109\right) y+253 c_p+83\right) y^5 x^6+\left(66 \left(59 c_p\right.\right.+254\Big) y^2-16 \left(253 c_p+83\right) y+935 c_p+10\right) y^4\right.\right.\right.\\ & \left.\left.\left. x^5-y^3 \left(2 (11 y (949 y-109)-670) y+11 \left(\left(738 y^2-988 y\right.\right.\right.+295\Big) y+20\Big) c_p+470\Big) x^4+\left(6 ((20 y (121 y\right.\right.\right.\right.\\ & -2)-913) y+408) y+(y ((20 (440 y-607) y+1407) y+2828)-576) c_p-286\Big) y^2 x^3-5 y \left(2 (y ((y (561 y\right.\\ & \left.+205)-794) y+513)-147) y+((y (2 (y (473 y-427)-644) y+1739)-609) y+77) c_p+30\right) x^2+10\\ & \left. \left(y \left((y ((11 y (9 y+20)-630) y+560)-251) y+(y ((y (99 y+235)\right.\right.-548) y+292)-56) (y-1) c_p+52\right)\\ & \left.\left.\left.\left.\left.-4\right) x-30 (y-1) \left(\left(((29 y-50) y+39) y+(y-1) ((29 y-25) y+7) c_p\right.\right.-14\right) y+2\right)\right) s m_s\right)F\left(s,x,y\right){}^2 \\ & +6 (y-1) s \left(7 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(22 \left(4 c_p+59\right) y^4 x^5\right.\right.-2 y^3 \Big(88 \left(3 c_p+23\right) y-143 c_p\\ & \left.\left.\left.+192\Big) x^4+\left(110 \left(11 c_p+39\right) y^2+20 \left(65-68 c_p\right) y+359 c_p-306\right) y^2 x^3-5 y \Big(44 \left(5 c_p+9\right) y^3+34\right.\right.\right.\\ & \left(9-7 c_p\right) y^2-2 \left(24 c_p+97\right) y+77 c_p+14\Big) x^2+10 \Big(y \Big(33 \left(c_p+1\right) y^3+4 \Big(c_p+22\Big) y^2-\left(93 c_p+86\right) y\\ & \left.\left. +56 c_p+12\Big)-1\Big) x-30 (y-1) \left(\left(7 \left(c_p+1\right) y-7 c_p-3\right) y+1\right)\Big) m_s m_c^2-s (x-1)x (y-1) y (y ((x ((x-3) x\right.\right.\\ & \left.\left.\left.\left.+3) y-3) y+3)-1) \left(\left(50 \left(2 c_p+65\right) y^3 x^4-140 y^2 \left(25 \left(c_p+3\right) y-18 c_p+1\right) x^3\right.\right.+7 \Big(150 \left(9 c_p+13\right) y^2\right.\right.\right.\right.\\ & \left.\left.\left.\left.\left.\left.-2 \left(851 c_p+361\right) y+483 c_p+79\Big) y x^2+70 \left(\left(-100 \left(c_p+1\right) y^2+7 \left(23 c_p+13\right) y\right.\right.-63 c_p-31\right) y+6\right) x\right.\right.\right.\right.\\ & \left.\left.\left.\left.\left.+70 (y-1) \left((25 y-21) \left(c_p+1\right) y+12\right)\right) y+70\right) m_c+21 (x-1) (y-1) \left(4 \left(4 c_p+59\right)\right.y^6 x^7+4 \Big(-327 y\right.\right.\right.\\ & \left.\left.\left.+(23-42 y) c_p+8\Big) y^5 x^6+\left(8 (381 y-32) y+2 \left(354 y^2-368 y+85\right) c_p+15\right) y^4 x^5-y^3 \Big((4 y (949 y\right.\right.\right.\\ & \left.-119)-195) y+2 \left(\left(738 y^2-988 y+295\right) y+20\right) c_p+90\Big) x^4+\left(3 ((4 y (220 y-7)-323) y\right.+168) y\right.\\ & \left.+2 (y ((4 (200 y-273) y+93) y+292)-64) c_p-68\right) y^2 x^3-5 y \left((y (4 (y (51 y+19)-78) y+231)-75) y\right.\\ & \left.+2 ((y (2 (y (43 y-35)-72) y+193)-75) y+11) c_p+9\right) x^2+10 \left(y \left((y ((2 y (9 y+22)-135) y+136)\right.\right.\\ & \left.\left.-70) y+2 (y ((y (9 y+25)-60) y+36)-8) (y-1) c_p+18\right)-2\right) x-30 (y-1) (3 (y-1) y+1) \left(2 (y-1)\right.\\ & \left.\left.\left.\left. \left(c_p+1\right) y+1\right)\right) s y m_s\right) F\left(s,x,y\right)+6 (y ((x ((x-3) x+3) y-3) y+3)-1) (y-1)^2 \left(7 \left(4 \left(4 c_p+59\right) y^4 x^5-4\right.\right.\right.\\ & \left.\left.\left. y^3 \left(8 \left(3 c_p+23\right) y-13 c_p+17\right) x^4+\left(4 (195 y+59) y+(4 (55 y-56) y+42) c_p-53\right) y^2 x^3-5 y \left(6 (2 y (6 y\right.\right.\right.\right.\\ & \left.+5)-7) y+(4 (y (10 y-7)-8) y+22) c_p+7\right) x^2+10 \left(6 \left(c_p+1\right) y^4+4 \left(2 c_p+5\right) y^3-5 \left(6 c_p+5\right) y^2+2 \right.\\ & \left.\left.\left.\left.\left(8 c_p+5\right) y-2\right) x-30 (y-1) \left(2 (y-1) \left(c_p+1\right) y+1\right)\right) m_c m_s-s (x-1) x (y-1) y^2 \left(4 \left(2 c_p+65\right) y^3 x^4\right.\right.\right.\\ & -56 y^2 \left(5 y \left(c_p+3\right)-4 c_p\right) x^3+7 \left(12 \left(9 c_p+13\right) y^2-4 \left(37 c_p+16\right) y+46 c_p+9\right) y x^2+35 \left(1-2 y\right.\\ \end{aligned}
\begin{aligned}[b] &\left.\left.\left.\left.\left. \left(8 (y-1) y\right.+2 (4 y-3) (y-1) c_p+3\right)\right) x+70 (y-1) \left(2 (y-1) \left(c_p+1\right) y+1\right)\right)\right) s^2 y m_c\right),\\ \rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{1,7;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{ \frac{c_1 m_c}{192 \pi ^3 (y-1)^3} F\left(s,x,y\right) \left(3 s (y-1) \left(y \left(x^2 \left(7 c_p-38\right)-7 c_p+44 x-4\right)+11 x y^4 \left(9 x^3\right.\right.\right. \\ & \left.\left.\left.\left(c_p-7\right)-12 x^2 \left(c_p-5\right)+2 x \left(c_p-11\right)+c_p+19 x^4+1\right)+y^3 \left(-(x-1) (x (11 x (9 x+5)-59)-7) c_p\right.\right.\right.\\ & \left.\left.\left.+2 ((12 x (x+4)-97) x+68) x-7\right)+(x-1) y^2 \left((11 x (8 x-5)-14) c_p-2 (6 (5 x-12) x+5)\right)-3 x+1\right)\right.\\ & \left. F\left(s,x,y\right)+2 \left((x-1) (y-1) c_p (x (y (2 x ((9 x-3) y-8)-2 y+9)-1)+y-1)+y \left(x \left(2 ((x ((19 x-63) x\right.\right.\right.\right.\\ & \left.\left.\left.\left.+60)-22) x+1) y^2+((2 x (x+7)-21) x+16) y-10 (x-3) x-22\right)-y\right)+x (4-3 x)+2 y-1\right) F(s,x,y)^2\right.\\ & \left.+6 s^2 (y-1)^2 y \left(2 x y^4 \left(9 x^3 \left(c_p-7\right)-12 x^2 \left(c_p-5\right)+2 x \left(c_p-11\right)+c_p+19 x^4+1\right)-2 y^3 \left((x (9 x-4)-1)\right.\right.\right. \\ & \left.(x-1) (x+1) c_p-2 ((x (x+5)-10) x+7) x+1\right)+(x-1) y^2 \left(2 (x (8 x-5)-2) c_p+x (31-12 x)-4\right)\\ & \left.\left.+2 x y \left(x \left(c_p-5\right)+7\right)-y \left(2 c_p+3\right)-2 x+1\right)\right)+\frac{c_2 x m_c}{128 \pi ^3 (x-1) (y-1)^4} F\left(s,x,y\right) \left(3 s (y-1) \left(-2 y^4 \left(\left(55 x^2\right.\right.\right.\right.\\ & \left.\left.-75 x+31\right) (x-1)^2 c_p+(x ((10 x-83) x+134)-103) x+31\right)+11 y^5 \left(((5 x-6) x+2) (x-1)^2 c_p\right.\\ & \left.\left.\left.+(x ((x ((7 x-30) x+51)-48) x+27)-10) x+2\right)+y^3 \left(((55 x-102) x+65) (x-1)^2 c_p-2 x ((2 x (x+17)\right.\right.\right.\\ & \left.\left.\left.-95) x+102)+75\right)+2 (x-1) y^2 \left((9 x-16) (x-1) c_p+(9 x-32) x+24\right)+x y \left(7 (x-2) c_p+11 x-26\right)\right.\right.\\ & \left.\left.+y \left(7 c_p+15\right)+2 x-2\right) F\left(s,x,y\right)+2 \left((x-1)^2 (y-1)^2 c_p (y (x (2 (5 x-6) y+3)+4 y-3)+1)+2 ((x ((x\right.\right. \\ & ((7 x-30) x+51)-48) x+27)-10) x+2) y^4+(x (((37-6 x) x-55) x+39)-11) y^3+(11-x ((2 x (x+4)\\ & \left.\left.-27) x+30)) y^2+(3 x-5) (x-1)^2 y+(x-1)^2\right) F\left(s,x,y\right){}^2+6 s^2 (y-1)^2 y \left(2 y^5 \left(((5 x-6) x+2) (x-1)^2 c_p\right.\right.\right.\\ & \left.+(x ((x ((7 x-30) x+51)-48) x+27)-10) x+2\right)-4 y^4 \left(((5 x-7) x+3) (x-1)^2 c_p+(x ((x-8) x+13)\right.\\ & -10) x+3\Big)+2 y^3 \left((5 (x-2) x+7) (x-1)^2 c_p+((21-8 x) x-22) x+8\right)+(x-1) y^2 \left(4 (x-2) (x-1) c_p\right.\\ & \left.\left.\left.+(4 x-15) x+12\right)+x y \left(2 (x-2) c_p+3 x-8\right)+y \left(2 c_p+5\right)+x-1\right)\right)\Bigg\},\\ \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{1,7;A(S)}(s) = &-\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1}{288 \pi ^3 (y-1)^2} \left(3 s (y-1) y F(s,x,y) \left(s (x-1) (y-1) \left(25 y^4 \left(x^4 \left(5 c_p-1\right)-4 x^3\right.\right.\right.\right.\\ & \left. \left(c_p+3\right)+6 x^2 \left(c_p+5\right)-8 x \left(c_p+1\right)+2 \left(c_p+1\right)\right)-2 y^3 \left(((x (42 x+115)-161) x+46) c_p+(x (152 x\right.\\ & \left.+185)-91) x+46\Big)+y^2 \left(21 ((7 x-6) x+2) c_p+(95 x-62) x+66\right)+12 (x-2) y+2\right)-6 x m_c^2 \left(2 c_p (y (x (4\right.\\ & \left.\left.\left. y-5)-2 y+2)+1)+2 y (4 (2 x-3) x y+x+6 y-4)+3\right)\right)+3 F\left(s,x,y\right){}^2 \left(s (x-1) (y-1) \left(35 y^4 \left(x^4 \Big(5 c_p\right.\right.\right.\right.\\ & \left.-1\Big)-4 x^3 \Big(c_p+3\Big)+6 x^2 \left(c_p+5\right)-8 x \left(c_p+1\right)+2 \left(c_p+1\right)\right)-y^3 \left(((x (96 x+295)-413) x+118) c_p+(x \right.\\ & \left.\left.(436 x+475)-233) x+118\right)+2 y^2 \left(12 ((7 x-6) x+2) c_p+(52 x-33) x+38\right)+12 (x-2) y+2\right)-3 x m_c^2\\ & \left.\left.\left. \left(2 y^2 \left(2 x \left(c_p-3\right)\right.\right.-c_p+4 x^2+3\right)-y \left((5 x-2) c_p+x+2\right)+c_p+1\right)\right)+2 (x-1) y \left(-4 y \left((x (x+4)-2)\right.\right.\\ & \left. (3 x-2) c_p+3 (6 x+5) x^2-9 x+4\right)+5 y^2 \Big(x^4 \left(5 c_p-1\right)-4 x^3 \left(c_p+3\right)+6 x^2 \left(c_p+5\right)-8 x \left(c_p+1\right)+2 \\ & \left.\left.\left(c_p+1\right)\Big)+3 ((7 x-6) x+2) c_p+6 x^2-3 x+6\right) F\left(s,x,y\right){}^3+3 s^3 (x-1) (y-1)^3 y^3 \left(2 x^4 y^3 \left(5 c_p-1\right)-8 x^3 \right.\right.\\ & y^2 (y+1) \left(c_p+3\right)+x^2 y \Big(12 y^2\left(c_p+5\right)-4 y \left(5 c_p+8\right)+14 c_p+9\Big)+2 x y \Big(-8 y^2 \left(c_p+1\right)+2 y \left(7 c_p+4\right)\\ & \left.\left. \left.-6 c_p-3\Big)+2 (y-1) \left(2 (y-1) y \Big(c_p\right.+1\Big)+1\right)+x\right)\right)+\frac{c_2}{128 \pi ^3 (y-1)^3} \left(-3 F(s,x,y)^2 \left(m_c^2 \left((y-1) c_p (x y-1)\right.\right.\right.\\ & \left. \left(\left(x^2+x-1\right) y-x+1\right)+y \left(x \left(\left(-x^2+x-3\right) y+(x ((8 x-15) x+9)-1) y^2-2 x+5\right)+y\right)-x-2 y+1\right)\\ & -s x (y-1) \left(y \left(35 x y^4\right.\right.\left(\left(x^3-2 x+1\right) c_p+(3 x ((x-5) x+8)-10) x+1\right)-y^3 \left((x (x (35 x+72)-13)-35)\right.\\ & \left.(x-1) c_p+(x ((11 x+115) x+275)-202) x+35\right)+y^2 \Big((x (37 x+46)-59) (x-1) c_p+5 (x (12 x+47)\\ & \left.\left.\left.-45) x+59\Big)-2 (x-1)^2 y \left(12 c_p+19\right)-11 x+12\right)-1\right)\right)+3 s (y-1) F(s,x,y) \left(s x (y-1) y \left(y \left(25 x y^4 \right.\right.\right.\\ \end{aligned}
\begin{aligned}[b]& \left.\left(\left(x^3-2 x+1\right) c_p+(3 x ((x\right.-5) x+8)-10) x+1\right)-y^3 \Big((x (x (25 x+63)-17)-25) (x-1) c_p+2 (x (x\\& \left.\left.\left.\left.+20) (2 x+5)-74) x+25\Big)+y^2 \left((x (38 x+29)-46) (x-1) c_p+(x (46 x+179)-175) x+46\right)-3 (x-1)^2\right.\right.\right.\right.\\ & \left.\left.\left. y \left(7 c_p+11\right)-11 x+12\right)-1\Big)-2 m_c^2 \left(y \left(2 (y-1) c_p (x y-1) \left(\left(x^2+x-1\right) y-x+1\right)+y \left(x \left(2 (x ((8 x-15)\right.\right.\right.\right.\right.\right.\\ & \left.\left.\left.\left.\left. x+9)-1) y^2-2 ((x-3) x+5) y-4 x+11\right)+2 y\right)-3 x-4 y+3\right)-1\right)\right)+2 x y \left((x-1) (y-1) c_p \left(x \Big(5 \Big(x^2\right.\right.\\ & \left.+x-1\Big) y^2-4 (x+1) y+3\Big)+5 y-3\right)+5 ((3 x ((x-5) x+8)-10) x+1) x y^3-(2 (x ((x+10) x+15)-12) x\\ & \left.\left.+5) y^2+((x (7 x+33)-30)x+8) y-3 (x-1)^2\right) F(s,x,y)^3+3 s^3 x (y-1)^3 y^3 \left(2 x y^4 \left(\left(x^3-2 x+1\right) c_p+(3 x\right.\right.\right.\\ & \left. ((x-5) x+8)-10) x+1\right)-2 y^3 \left((x (x+2)-4) x^2 c_p+c_p+2 (2 x (x+2)-3) x+1\right)+y^2 \left(2 \left(2 x^2+x-2\right)\right.\\ & \left.\left.\left. (x-1) c_p+(x (4 x+15)-15) x+4\right)-(x-1)^2 y \left(2 c_p+3\right)-x+1\right)\right)\Bigg\},\\ \rho^{\langle\bar{s}s\bar{s}s\rangle}_{1,7;A(S)}(s) =&- \int^{z_{\max}}_{z_{\min}}{\rm d}z \frac{c_1}{24 \pi } \left(m_c m_s \left(s (z-1) \left(z \left(-14 c_p+22 z-5\right)+2\right) G(s,z)+\left(-2 c_p+4 z-3\right) G(s,z)^2+2 s^2 (z-1)^2 z\right.\right.\\ & \left.\left. \left(z \left(-2 c_p+2 z-1\right)+1\right)\right)+m_s^2 \left(-2 s \left(z \left(z \left((35 z-24) (z-1) c_p+(35 z-59) z+38\right)-12\right)+1\right) G(s,z)\right.\right.\\ & \left.\left.-2 z (5 z-3) \left(c_p+1\right) G(s,z)^2+s^2 (-(z-1)) z \left((z-1) z \left((25 z-21) z \left(c_p+1\right)+12\right)+1\right)\right)+2 m_c^2 G(s,z)\right.\\& \left. \left(c_p G(s,z)+G(s,z)+4 s (z-1) z \left(c_p+1\right)+2 s\right)\right)+\int^{1}_{0}{\rm d}z \frac{c_1}{24 \pi }s^3 (z-1)^3 z^3 m_s^2 \left(2 (z-1) z \left(c_p+1\right)+1\right),\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{1,7;A(S)}(s) =& - \int^{z_{\max}}_{z_{\min}}{\rm d}z\Bigg\{\frac{ c_1}{288 \pi } \left(12 m_c^2 \left(\left(\left(8 z^2-4 z-2\right) c_p+8 z^2-4 z+2\right) G(s,z)+2 s (z (11 z-7)-2) (z-1) z c_p\right.\right.\\ & \left.\left.+2 s (z ((11 z-18) z+14)-5) z+s\right)+m_c m_s \left(s \left(-4 (z-1) (10 z (12 z-7)-21) z c_p+((5 ((10 z-59) z+9) z\right.\right.\right.\\ & \left.\left.\left.+51) z+45) z-12\right)-8 \left((z (15 z-7)-3) c_p+3 (5 z-2) z+3\right) G(s,z)\right)-4 z m_s^2 \left(2 \left(2 (2 (5 (3 z-4) z+4) z\right.\right.\right.\\ & \left.+3) c_p+4 (5 (3 z-4) z+9) z-9\right) G(s,z)+s (z-1) \left(((2 (z (130 z-77)-115) z+127) z+8) c_p+2 (z ((130 z\right.\\ & \left.\left.\left.-77) z+11)-10) z\right)\right)\right)+\frac{c_2 }{64 \pi }\left(4 m_c^2 \left(\left(c_p+1\right) G(s,z)+2 s (z-1) z c_p+2 s (z-1) z+s\right)+m_c m_s \left(\Big(-4 c_p\right.\right.\\ & \left.\left.+8 z-6\Big) G(s,z)+s (z-1) \left(-14 z c_p+(22 z-5) z+2\right)\right)-2 m_s^2 \left(2 (5 z-3) z \left(c_p+1\right) G(s,z)+s z \Big(z \Big((35 z\right.\right.\\ & \left.\left.-24) (z-1) c_p+(35 z-59) z+38\Big)-12\Big)+s\right)\right)\Bigg\}+\int^{1}_{0}{\rm d}z\Bigg\{\frac{c_1 s^2 (z-1) z}{144 \pi } \left(m_c m_s \left(z \left(3 (z (35 z-27)-4)\right.\right.\right.\\ & \left.\left.+\left(z-1) c_p(60 z^2-58 z+15\right) z-4\right)-2\right)+2 (z-1) z m_s^2 \Big((((2 z (73 z-77)-39) z+37) z+12) c_p\\ & \left.+(z (2 (73 z-77) z+79)-26) z-4\Big)-12 (z-1) z m_c^2 \left(2 (z-1) z \left(c_p+1\right)+1\right)\right)-\frac{{\rm i} c_2 s^2 (z-1)^2 z}{32 \pi } m_c m_s\\ & \left(z \left(-2 c_p+2 z-1\right)+1\right)\Bigg\},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{1,7;A(S)}(s) = & \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{x c_1}{576 \pi ^3 (x-1)^3 (y-1)^4} m_c \left(-3 (x-1) \left(2 ((x ((x ((7 x-37) x+81)-95) x+61)-21) x\right.\right.\\ & +3) x y^5-((x ((6 (x-4) x+13) x+9)-15) x+5) y^4+((x ((-2 (x-4) x-43) x+66)-49) x+14) y^3\\ & +(x ((17 x-42) x+41)-14) y^2-3 (x-2) (x-1)^2 y-(x-1)^2+(y ((y (-2 x+2 (4 x-3) (x-1) y+7)-3) x\\ & \left.-5 y+4)-1) (x-1)^2 (y-1)^2 c_p\right) F(s,x,y)^2+\left(-2 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(2 ((x ((7 x-23) x\right.\right.\\ & +24)-10) x+1) x y^3+(x ((5 x-9) x+7)-1) y^2-(x-1) ((5 x-9) x+2) y-(x-1)^2+(x-1) \\ & \left.\left(2 \left(x^2+x-1\right) x y^2+x (2-5 x) y+y+x-1\right) (y-1) c_p\right) m_c^2-3 s (x-1) (y-1) \left(11 \left((4 x-3) c_p (x-1)^3\right.\right.\\ & \left.+(x ((x ((7 x-37) x+81)-95) x+61)-21) x+3\right) x y^6+\Big(-(11 x (8 x-5)-29) c_p (x-1)^3+2 ((2 x (2\\ & -5 (x-4) x)-53) x+50) x-29\Big) y^5+\Big(((11 x (4 x-5)-65) x+83) c_p (x-1)^2-4 x ((x ((x-4) x+48)\\ & -83) x+66)+79\Big) y^4+\left(-((11 x-76) x+86) c_p (x-1)^2+2 (9 (5 x-13) x+122) x-89\right) y^3-(x-1) \\ \end{aligned}
\begin{aligned}[b] & \left.\left.\left((17 x-64) x+3 (6 x-13) (x-1) c_p+53\right) y^2-\left(7 c_p+16\right) y+\left(-10 x-7 (x-2) c_p+26\right) x y-2 x+2\right)\right)\\ & F(s,x,y)-s (y-1) \left((y ((x ((x-3) x+3) y-3) y+3)-1) \left(11 \left(7 x^4+\left(c_p-23\right) x^3+24 x^2-2 \left(c_p+5\right) x\right.\right.\right.\\ & \left.+c_p+1\right) x y^4-\left(-2 ((21 x-40) x+28) x+(x (x (11 x+37)-19)-7) (x-1) c_p+7\right) y^3+\left(-22 x^2+50 x\right.\\ & \left.\left.+(x (26 x-1)-14) c_p-10\right) (x-1) y^2-(x-1) \left(10 x+7 (x-1) c_p-4\right) y-x+1\right) m_c^2+3 (x-1) (y-1)\\& \left(2 \left((4 x-3) c_p (x-1)^3+(x ((x ((7 x-37) x+81)-95) x+61)-21) x+3\right) x y^6-2 \left((8 x+3) c_p (x-1)^4\right.\right.\\ & \left.+2 \left(x^4-4 x^3+5 x-5\right) x+3\right) y^5+\left(2 ((x (4 x-5)-7) x+9) c_p (x-1)^2-3 x (11 (x-2) x+19)+18\right) y^4\\ & +\left(-2 ((x-8) x+10) c_p (x-1)^2+((19 x-54) x+60) x-23\right) y^3-(x-1) \left((4 x-17) x+2 (2 x-5) (x-1)\right.\\ & \left.\left.\left.\left. c_p+16\right) y^2-2 \left(c_p+3\right) y+\left(-3 x-2 (x-2) c_p+9\right) x y-x+1\right) s y\right)\right)\Bigg\}\\ & -\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s^2 x y}{576 \pi ^3 (x-1)^3 (y-1)^2} m_c^3 (y (y (x ((x-3) x+3) y-3)+3)-1) \left(-2 y^3 \left((x (x+2)-4) x^2 c_p\right.\right.\\ & \left.\left.+c_p-2 (2 (x-2) x+3) x+1\right)+(x-1) y^2 \left(2 \left(2 x^2+x-2\right) c_p+x (11-4 x)-4\right)+2 x y^4 \left(x^3 \left(c_p-23\right)\right.\right.\\ & \left.\left.-2 x \left(c_p+5\right)+c_p+7 x^4+24 x^2+1\right)-(x-1)^2 y \left(2 c_p+3\right)-x+1\right),\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{1,7;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1}{576 \pi ^3 (x-1)^3 (y-1)^3} m_c^2 \left(-(y ((x ((x-3) x+3) y-3) y+3)-1) \left(5 c_p x^2+x^2-6 c_p\right.\right.\\ & x-2 x+\left(\left(c_p+9\right) x^4-4 \left(c_p+5\right) x^3+2 \left(5 c_p+9\right) x^2-8 \left(c_p+1\right) x+2 \left(c_p+1\right)\right) y^2+2 \left(((x-5) x+5) x\right.\\ & \left.\left.+(x ((x-7) x+7)-2) c_p-2\right) y+2 c_p+2\right) m_c^2+(x-1) (y-1) \left(35 \left((3 x ((x-4) x+6)-8) x+((x ((x-4)\right.\right.\\ & \left. x+10)-8) x+2) c_p+2\right) ((x-3) x+3) x^2 y^7+\left(((x (x (x (10-81 x)+374)-256)-837) x+486) x\right.\\ & \left.+(x (((x (59 (x-10) x+1724)-2176) x+543) x+486)-210) c_p-210\right) x y^6+\left(((x (((393-26 x) x\right.\\ & \left.-1115) x+2667)-1635) x+684) x+3 (x ((x (((38 x-121) x+41) x+643)-721) x+228)-8) c_p-24\right)\\ &y^5+\Big((x (1537-3 x ((35 x-99) x+571))-892) x+((x (((437-47 x) x-2273) x+2431)-880) x+72) c_p\\ & +72\Big) y^4+\left(((x (20 x+491)-737) x+634) x+(x (((917-95 x) x-1217) x+574)-84) c_p-96\right) y^3\\ & +\left(((180-43 x) x-256) x-12 (x-1) ((11 x-13) x+4) c_p+72\right) y^2-6 \left(2 c_p+5\right) y+3 \Big(-7 x+2 (6-5 x)\\& \left. c_p+18\Big) x y-5 x+6\right) s+(x-1) \left(10 \left((3 x ((x-4) x+6)-8) x+((x ((x-4) x+10)-8) x+2) c_p+2\right)\right.\\ & ((x-3) x+3) x^2 y^6+2 \left(\left(72-x \left(\left(12 x^3-63 x+52\right) x+114\right)\right) x+(x (((x (8 (x-10) x+233)-292) x+66)\right.\\ & \left. x+72)-30) c_p-30\right) x y^5+3 \left(((3 x (((22-3 x) x-54) x+106)-194) x+72) x+(x ((x ((x (9 x-26)-6) x\right.\\ & \left.+200)-218) x+72)-4) c_p-4\right) y^4+2 \left((x (277-3 x (6 (x-4) x+103))-142) x+2 ((x (((37-4 x) x\right.\\ & \left.-172) x+191)-77) x+9) c_p+18\right) y^3-2 \left((3 x ((x-26) x+38)-88) x+((x ((17 x-137) x+203)-112)\right.\\ & \left.x+21) c_p+21\right) y^2-6 \left(((x-7) x+9) x+(3 x (2 (x-3) x+5)-4) c_p-4\right) y-3 \left((x-2) x+((5 x-6) x+2)\right.\\ & \left.\left.\left. c_p+2\right)\right) F(s,x,y)\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{ c_1 s}{1152 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1)\\ & \left(2 m_c^2 \left(2 y^3 \left(x^4 \left(c_p+9\right)-4 x^3 \left(c_p+5\right)+2 x^2 \left(5 c_p+9\right)-8 x \left(c_p+1\right)+2 \left(c_p+1\right)\right)+4 y^2 \left((x ((x-7) x+7)\right.\right.\right.\\ & \left.\left.\left.-2) c_p+x (x-2)^2-2\right)+x y \left(2 (5 x-6) c_p+3 (x-2)\right)+y \left(4 c_p+6\right)+x-2\right)-s (x-1) x (y-1) y \left(25 y^4\right.\right.\\ & \left(((x ((x-4) x+10)-8) x+2) c_p+(3 x ((x-4) x+6)-8) x+2\right)+2 y^3 \Big(23 (x ((x-7) x+7)-2) c_p+(91\\ & \left.\left.-x (27 x+91)) x-46\Big)+y^2 \left(21 ((5 x-6) x+2) c_p+(29 x-62) x+66\right)+12 (x-2) y+2\right)\right)\Bigg\},\\ \rho^{\langle g_sG^2\rangle^2}_{1,7;A(S)}(s) = &-\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^3 y^4}{5806080 \pi ^5 (x-1)^2 (y-1)^2} m_c^4 \left(28 y \Big((x ((15 x-74) x+70)-20) c_p-39 x^2+45 x\right.\\ \end{aligned}
\begin{aligned}[b]& \left.-20\Big)+10 y^2 \left(x^4 \left(2 c_p+65\right)-70 x^3 \left(c_p+3\right)+21 x^2 \left(9 c_p+13\right)-140 x \left(c_p+1\right)+35 \left(c_p+1\right)\right)+21 \Big(((23 x\right.\\& \left.-30) x+10) c_p-(2 x+5) x+10\Big)\right)-\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s x^3 y^3}{1658880 \pi ^5 (x-1)^2 (y-1)} m_c^4 \Big(y^2 \left(8 x^2 \left(69 c_p+8\right)\right.\\ & \left.-30 x \left(24 c_p+11\right)+240 c_p+380\right)+y^3 \left(50 x^3 \left(9 c_p-1\right)-x^2 \left(2183 c_p+923\right)+5 x \left(413 c_p+233\right)\right.\\ & \left.-590 \left(c_p+1\right)\right)+10 y^4 \left(x^4 \left(2 c_p+65\right)-70 x^3 \left(c_p+3\right)+21 x^2 \left(9 c_p+13\right)-140 x \left(c_p+1\right)+35 \left(c_p+1\right)\right)\\ & +60 (x-2) y+10\Big),\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{1,7;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1}{2304 \pi ^3 (x-1)^3 (y-1)^3} m_c \left(6 (x-1) \left(y^4 \left(\left(2 \left(8 x^3-15 x+6\right) x+5\right) (x-1)^2 c_p\right.\right.\right.\\ & \left.+(x (2 (3 x ((x-2) x-4)-7) x+33)-24) x+5\right)-2 x y^5 \left(\left(8 x^2-3\right) (x-1)^3 c_p+(x ((x ((19 x-97) x+201)\right.\\ & \left.-219) x+121)-33) x+3\right)+2 y^3 \left(-(x-1) (x (((8 x-19) x+12) x+5)-7) c_p+\left(\left(2 x \left(x^2+x+19\right)-53\right)\right.\right.\\& \left.\left.\left.\left. x+33\right) x-7\right)+2 y^2 \left((x ((x-3) x+6)-7) (x-1) c_p-((x (4 x+13)-33) x+28) x+7\right)+6 (x-1) y \Big(c_p\right.\right.\\ & \left.+(x-3) x+1\Big)+(1-x) \left(x \left(c_p-3\right)+c_p+1\right)\right) F(s,x,y)-2 m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \Big((x\\ & -1)(y-1) c_p (x (y (2 x ((9 x-3) y-8)-2 y+9)-1)+y-1)+y \left(x \left(2 ((x ((19 x-63) x+60)-22) x+1) y^2\right.\right.\\ & \left.\left.+((2 x (x+7)-21) x+16) y-10 (x-3) x-22\right)-y\right)+x (4-3 x)+2 y-1\Big)+3 s (x-1) (y-1) \left(y^5 \Big(\Big(11\right.\\ & \left(8 x^3-15 x+6\right) x+29\Big) (x-1)^2 c_p+2 (x ((x (10 (x-2) x-53)-100) x+156)-90) x+29\Big)-11 x y^6\\& \left(\left(8 x^2-3\right) (x-1)^3 c_p+(x ((x ((19 x-97) x+201)-219) x+121)-33) x+3\right)+y^4 \left(-(x-1) (x (11 ((8 x\right.\\ & \left.-19) x+12) x+58)-83) c_p+2 ((x (4 (x+8) x+213)-340) x+213) x-79\right)+y^3 \left((11 x ((x-3) x+6)\right.\\ & \left.-86) (x-1) c_p-2 (9 (2 x (x+6)-29) x+208) x+89\right)+y^2 \Big(3 (x-1) (x+13) c_p+((51 x-209) x+217) x\\ & \left.\left.-53\Big)+x y \left(x \left(30-7 c_p\right)-52\right)+y \left(7 c_p+16\right)+4 x-2\right)\right)-\frac{c_2 x y^2}{768 \pi ^3 (x-1)^2 (y-1)^2} m_c^3 \Big(y \Big(((x (x+21)\\ & -21) x+5) c_p+\left(x^2+x-1\right) x+5\Big)-2 \left((1-2 x)^2 c_p-x^2+x+1\right)+2 x y^3 \left((x-1) ((x-4) x+6) x c_p+c_p\right.\\ & \left.+(3 (x-2) x+2) x+1\right)+y^2 \left(3 (x (((x-4) x-2) x+4)-1) c_p+(((20-11 x) x-12) x+2) x-3\right)\Big)\Bigg\}\\ & +\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s }{2304 \pi ^3 (x-1)^3 (y-1)^2}m_c \left(m_c^2 (-(y (y (x ((x-3) x+3) y-3)+3)-1)) \left(y \left(x^2 \left(7 c_p-38\right)\right.\right.\right.\\ & \left.-7 c_p+44 x-4\right)+11 x y^4 \left(9 x^3 \left(c_p-7\right)-12 x^2 \left(c_p-5\right)+2 x \left(c_p-11\right)+c_p+19 x^4+1\right)+y^3 \left(-(x-1)\right. \\& \left.(x (11 x (9 x+5)-59)-7) c_p+2 ((12 x (x+4)-97) x+68) x-7\right)+(x-1) y^2 \left((11 x (8 x-5)-14) c_p\right.\\ & \left.\left.-2 (6 (5 x-12) x+5)\right)-3 x+1\right)-3 s (x-1) (y-1) y \left(2 y^5 \left(-\left(\left(8 x^3-15 x+6\right) x+3\right) (x-1)^2 c_p+2 ((x (x\right.\right.\\ & \left.\left. (5-(x-2) x)+9)-15) x+9) x-3\right)+2 x y^6 \left(\left(8 x^2-3\right) (x-1)^3 c_p+(x ((x ((19 x-97) x+201)-219) x\right.\right.\\ & \left.+121)-33) x+3\right)+y^4 \left(2 (x (((8 x-19) x+12) x+6)-9) (x-1) c_p-3 ((x (6 x+25)-46) x+31) x+18\right)\\ & +y^3 \left(-2 (x-1) (x ((x-3) x+6)-10) c_p+((x (7 x+46)-123) x+104) x-23\right)+y^2 \left(-2 (x-1) (x+5) c_p\right.\\ & \left.\left.\left.-(x-2) (12 x-31) x+16\right)+x y \left(x \left(2 c_p-9\right)+18\right)-2 y \left(c_p+3\right)-2 x+1\right)\right)-\frac{c_2 s x y }{1536 \pi ^3 (x-1)^2 (y-1)}\\ & m_c^3 \left(y \left(-y \left(11 (1-2 x)^2 c_p-6 x^2+8 x+24\right)+11 x y^4 \left((x-1) ((x-4) x+6) x c_p+c_p+(3 (x-2) x+2) x+1\right)\right.\right.\\ & +y^3 \left(3 (2 x ((3 (x-4) x-4) x+10)-5) c_p-(((59 x-104) x+44) x+8) x-15\right)+y^2 \left(((x (7 x+111)-111)\right.\\ & \left.\left.\left. x+26) c_p+2 ((5 x-7) x+7) x+30\right)+9\right)-1\right)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{1,7;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 }{3456 \pi ^3 (x-1)^2 (y-1)^2}m_c^2 \left(10 ((x-3) x+3) x y^6 \left(x^4 \left(5 c_p-1\right)-4 x^3 \left(c_p+3\right)+6 x^2\right.\right.\\ \end{aligned}
\begin{aligned}[b]& \left. \left(c_p+5\right)-8 x \left(c_p+1\right)+2 \left(c_p+1\right)\right)-2 y^5 \left(12 x^6 \left(c_p+6\right)+4 x^5 \left(c_p-39\right)-5 x^4 \left(13 c_p+3\right)+4 x^3 \Big(61 c_p\right.\\ & \left.+31\Big)+42 x^2 \left(7-3 c_p\right)-72 x \left(c_p+1\right)+30 \left(c_p+1\right)\right)+6 y^4 \left(((x (((7 x-8) x+48) x+28)-84) x+26) c_p\right.\\ & \left.+(x (255-2 x (5 (x-3) x+23))-88) x+26\right)-y^3 \left(((x (5 (x+28) x+228)-452) x+152) c_p+(x (x (168\right.\\ & \left.-19 x)+930)-458) x+152\right)+y^2 \left(((x (15 x+98)-184) x+68) c_p+3 (x (21 x+88)-66) x+68\right)\\ & \left.+3 y \left((x (x+12)-4) c_p-(x-14) x-4\right)-9 x \left(c_p+1\right)\right)+\frac{c_2 y}{1536 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \left(3 y \left((x ((7 x-17) x\right.\right.\\ & \left.+11)-2) c_p+\left(-3 x^2+x+3\right) x-2\right)+10 x y^5 \left(((x ((x ((x-6) x+15)-20) x+19)-10) x+2) c_p\right.\\ & \left.+(x ((9 x-19) x+8)-2) (x-1)\right)+4 x y^4 \left((4 x ((x ((x-5) x+10)-19) x+14)-13) c_p+(46-x ((x (5 x\right.\\ & \left.+17)-90) x+106)) x-13\right)+x y^3 \Big(3 ((x ((x-3) x+66)-66) x+17) c_p+(x ((33 x-145) x+216)-112)\\ & x+51\Big)+y^2 \left((3-2 x ((x (3 x+44)-56) x+20)) c_p+2 (x ((7 x-18) x+12)-14) x+3\right)\\ & \left.+3 \left((1-2 x)^2 c_p+1\right)\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{c_1 }{3456 \pi ^3 (x-1)^3 (y-1)^2}m_c^2 \left(s (x-1) (y-1) \left(y \Big(35 ((x-3) x\right.\right.\\ & +3) x y^6 \left(x^4 \left(5 c_p-1\right)-4 x^3 \left(c_p+3\right)+6 x^2 \left(c_p+5\right)-8 x \left(c_p+1\right)+2 \left(c_p+1\right)\right)-y^5 \left(4 x^6 \left(24 c_p+109\right)\right.\\ & +7 x^5 \left(c_p-119\right)-5 x^4 \left(97 c_p+91\right)+2 x^3 \left(911 c_p+491\right)+9 x^2 \left(233-107 c_p\right)-486 x \left(c_p+1\right)+210\\ & \left. \left(c_p+1\right)\right)+y^4 \left(3 ((x (((56 x-65) x+344) x+241)-621) x+188) c_p+(x (((21-40 x) x-194) x+4797)\right.\\ & \left.-1527) x+564\right)-y^3 \left(((x ((85 x+364) x+1203)-1807) x+568) c_p+(x (x (552-29 x)+3291)-1465)\right.\\ & \left.x+652\right)+y^2 \left(((x (78 x+583)-773) x+262) c_p+(x (267 x+1087)-767) x+418\right)-2 y \left(3 ((13 x-24)\right.\\ & \left.\left.x+8) c_p+(61 x-114) x+77\right)-3 x \left(6 c_p+13\right)+30\Big)-2\right)-3 x m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1)\\ & \left.\left(2 y^2 \left(2 x \left(c_p-3\right)-c_p+4 x^2+3\right)-y \left((5 x-2) c_p+x+2\right)+c_p+1\right)\right)+\frac{c_2 y}{1536 \pi ^3 (x-1)^3 (y-1)^2} m_c^2\\ & \left(m_c^2 \left(4 x^2 c_p+y^4 \left(((x ((x ((x-6) x+15)-20) x+19)-10) x+2) c_p+(x ((x ((x-6) x+23)-36) x+27)\right.\right.\right.\\ & \left.-10) x+2\right)+2 y^3 \left((x ((x ((x-5) x+10)-18) x+13)-3) c_p+((x ((x-13) x+26)-24) x+11) x-3\right)\\ & +y^2 \left(((x ((x-4) x+30)-28) x+7) c_p+(x ((9 x-20) x+26)-16) x+7\right)-4 y \Big((1-2 x)^2 c_p+(x-1) x\\ & \left.+1\Big)-4 x c_p+c_p+1\right)-s (x-1) (y-1) \left(35 x y^6 \left(((x ((x ((x-6) x+15)-20) x+19)-10) x+2) c_p\right.\right.\\ & \left.+(x ((9 x-19) x+8)-2) (x-1)\right)+x y^5 \left((x ((59 x ((x-5) x+10)-1106) x+811)-188) c_p+(631\right.\\ & \left.-x ((x (70 x+247)-1290) x+1486)) x-188\right)+x y^4 \left(3 (2 (x ((3 x-11) x+132)-128) x+65) c_p\right.\\ & \left.+(x ((137 x-568) x+786)-406) x+223\right)+y^3 \left((6-x (4 (x (3 x+95)-107) x+131)) c_p+(x ((37 x\right.\\& \left.-120) x+84)-147) x+6\right)+y^2 \left(6 (3 x-2) (5 (x-1) x+1) c_p+(x (23-18 x)+49) x-12\right)\\ & \left.\left.\left. +x y \left(24 (x-1)\right.c_p-3 x-8\right)+y \left(6 c_p+9\right)+x-3\right)\right)\Bigg\},\\ \end{aligned}
(23) where
F(s,x,y)=\dfrac{m_c^2 (1-x y)}{1-x}-s (1-y) y ,G(s,z)=m_c^2-s(1-z)z ,y_{\max}=\dfrac{1}{2}+\dfrac{\sqrt{4 m_c^2 s (x-1)+\left(s (x-1)-m_c^2 x\right)^2}+m_c^2 x}{2 s (1-x)} ,y_{\min}=\dfrac{1}{2}-\dfrac{\sqrt{4 m_c^2 s (x-1)+\left(s (x-1)-m_c^2 x\right)^2}-m_c^2 x}{2 s (1-x)} ,x_{\max}=\left(1-2 \sqrt{m_c^2/s}\right)/\left(\sqrt{m_c^2/s}-1\right)^2 ,z_{\max}=\dfrac{1}{2}\left(1+\sqrt{1-4 m_c^2/s}\right) ,z_{\min}=\dfrac{1}{2}\left(1-\sqrt{1-4 m_c^2/s}\right) , coefficientc_p=1 for currentJ_{1,\mu\nu}^{A(S)} whilec_p=-1 for currentJ_{7,\mu\nu}^{A(S)} , andc_1=12,c_2=-8,c_3=4 for color antisymmetric currentJ_{i,\mu\nu}^{A} whilec_1=24,c_2=8,c_3=20 for color symmetric currentJ_{i,\mu\nu}^{S} . The spectral functions for(1,1\{1,0\}) structure are shown as\begin{aligned}[b]\rho^{\rm pert}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{x^2 }{51609600 \pi ^5 (y-1)^5}F(s,x,y)^3 c_1 \left((x-1) x \left((350 (4 y-3) y+(231-8 y (-266 x+5 ((x\right.\right.\\ & -14) x+42) y+147)) x y+35) x+(x (140 (4 y-1) y+(8 (5 x (20 x y+21)-28 (5 y+4)) y+483) x y-105)\\& \left.-70 (y-1)) c_p\right) F(s,x,y)^3+42 x \left((x-1) (y-1) s \left(\left(\left(\left(-20 ((x-14) x+42) y^2+6 (141 x-32) y-29\right) x\right.\right.\right.\right.\\ & \left.\left.+500 y-410\right) y+105\right) x y+\left(-20 (y-1) y+2 \Big(55 (2 y-1) y+\left(\left(115 x+40 \left(5 x^2-7\right) y-96\right) y+143\right) x y\right.\\ & \left.\left.-15\Big) x y-5\right) c_p\right)+\left(50 (y-1)+\left(-10 (3 y+10) y+((8 (7 x-8) y x-183 x+166) y+140) x y+2 (y (x (y (11 x\right.\right.\\ & \left.\left.\left.+8 ((x-9) x+5) y+68)-20)-40 y)+5) c_p+15\right) x\right) m_c m_s\right) F(s,x,y)^2+30 (y-1) \Big((x-1) (y-1) \\ & \left(50 \left(-(x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^3+14 \left((131 x+3) x+4 ((5 x+2) x+5) c_p+75\right) y^2+7 \left((82 x-30)\right.\right.\\ & \left.\left. c_p-35 (x+4)\right) y+385\right) s^2 x^2 y^2+14 \left(30 (y-1)+\left(\left(-2 \left(7 c_p+17\right) y x^2+5 \left(8 c_p y+12 y-2 c_p+3\right) x-20 y\right.\right.\right.\\ & \left.\left.\left. \left(c_p+2\right)-10\right) y+5\right) x\right) m_c^2 m_s^2+7 \left(10 (y+1) y+(-30 (y+4) y+((22 (7 x-8) y x-419 x+330) y+346) x y\right.\\ & \left.-125) x y+2 (y ((((2 y (34 x+11 ((x-9) x+5) y+75)-107) x-110 y+30) y+30) x+10 y)-5) c_p+20\right) s x\\ & \left.m_c m_s\Big) F(s,x,y)+120 (y-1)^2 s \left((x-1) (y-1) \left(\left(4 \left(-(x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^2+14 \left(\Big(9 x+4 c_p\right.\right.\right.\right.\right.\\ & \left.\left.\left.+3\Big) x+5\right) y+7 \left(4 c_p-5\right) x-70\right) y+35\right) s^2 x^2 y^3+7 \left(10 (y-1) y+\left(y (x ((4 (7 x-8) y x-77 x+82) y+44)\right.\right.\\ & \left.\left.-30 y)+2 (2 y (9 x+2 ((x-9) x+5) y-5)-5) (x y-1) c_p-25\right) x y+10\right) s x m_c m_s y+14 \left(\left(-4 \left((17 x-30) x\right.\right.\right.\\ & \left.\left.\left.\left.\left.+((7 x-20) x+10) c_p+20\right) x y^2+5 \left(\left(x+(4-8 x) c_p+6\right) x+6\right) y+10 \left(c_p-2\right) x-30\right) y+15\right) m_c^2 m_s^2\right)\right),\\ \rho^{\langle\bar{s}s\rangle}_{2,8;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 (x-1) x^2}{3072 \pi ^3 (y-1)^3} m_c F(s,x,y)^2 \Big(4 s (y-1) y (11 x y-5) (2 x y-1) F(s,x,y)+(x y-1)\\& (8 x y+1) F(s,x,y)^2+12 s^2 (y-1)^2 y^2 (x y (4 x y-3)+1)\Big),\\ \rho^{\langle m_s\bar{s}s\rangle}_{2,8;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x}{6144 \pi ^3 (y-1)^3} F(s,x,y) \left(12 s (y-1) F(s,x,y) \left(2 m_c^2 \Big(x y \left(-4 ((x-2) x+2) y^2+6 y-3\right)\right.\right.\\ & \left.+2 (y-1) y+1\Big)+s (x-1) x (y-1) y^2 \left(y \left(50 ((x-2) x+2) x y^2-2 (x+15) (2 x+1) y+29 x+28\right)-11\right)\right)\\ & +4 F(s,x,y)^2 \left(4 m_c^2 (y (-((x-2) x+2) x y+x+1)-1)+s (x-1) x (y-1) y (y (x (2 y (70 ((x-2) x+2) y-11 x\right.\\ & \left.-68)+57)-100 y+82)-21)\right)+(x-1) c_p \left((x (y (-8 ((x-4) x+2) y-15 x+4)+3)+2 (y-1)) F(s,x,y)^2\right.\\& -4 s (y-1) (2 y (x (y (11 ((x-4) x+2) y+25 x-11)-3)-2 y+2)-1) F(s,x,y)-24 s^2 x (y-1)^2 y^3 (2 ((x-4) x\\ & \left.+2) y+5 x-3)\right) F(s,x,y)+(x-1) x \left(40 ((x-2) x+2) x y^3-8 ((x+3) x+5) y^2+3 (x+10) y-1\right) F(s,x,y)^3\\ & \left.+24 s^3 (x-1) x (y-1)^3 y^3 \left(y \left(x \left(4 ((x-2) x+2) y^2-6 y+3\right)-2 y+2\right)-1\right)\right),\\ \rho^{\langle g_sG^2\rangle}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{x^2 c_1}{30965760 \pi ^5 (x-1)^3 (y-1)^5} m_c \left((x-1) x \left((-350 (4 y-3) y+(8 y (-266 x+5 ((x-14) x\right.\right.\\ & +42) y+147)-231) x y-35) (x y-1) ((x+((x-3) x+3) y-3) y+1) (-x) m_c+21 \left(8 ((x ((7 x-41) x+101)\right.\\& -108) x+40) x^2 y^5+(x (((23-31 x) x-469) x+1056)-610) x y^4+(((x (25 x+541)-1020) x+320) x\\ & \left.+310) y^3+(13 x (x (5-14 x)+35)-600) y^2+5 ((23 x-65) x+74) y+45 x-80\right) m_s+c_p \left((x y-1) ((x+((x\right.\\ & -3) x+3) y-3) y+1) (x (140 (4 y-1) y+(8 (5 x (20 x y+21)-28 (5 y+4)) y+483) x y-105)-70 (y-1)) m_c\\ & +42 \left(8 ((x ((x-13) x+38)-44) x+15) x^2 y^5+(x ((x (27 x-71)-7) x+278)-140) x y^4+((((103-15 x) x\right.\\ & \left.\left.\left.-330) x+100) x+20) y^3+(x (x (95-16 x)+35)-30) y^2-5 (x (x+7)-2) y+5 x\right) m_s\right)\right) F(s,x,y)^3\\ & +21 \left((y ((x ((x-3) x+3) y-3) y+3)-1) \left(50 (y-1)+\left(-10 (3 y+10) y+((8 (7 x-8) y x-183 x+166) y\right.\right.\right.\\ & \left.\left.+140) x y+2 (y (x (y (11 x+8 ((x-9) x+5) y+68)-20)-40 y)+5) c_p+15\right) x\right) x m_s m_c^2+(x-1) (y-1)\\ & \left(s x (y ((x ((x-3) x+3) y-3) y+3)-1) \left(\Big(\left(\left(-20 ((x-14) x+42) y^2+6 (141 x-32) y-29\right) x+500 y-410\right) y\right.\right.\\ & \left.+105\Big) x y+\left(-20 (y-1) y+2 \left(55 (2 y-1) y+\left(\left(115 x+40 \left(5 x^2-7\right) y-96\right) y+143\right) x y-15\right) x y-5\right) c_p\right)\\ \end{aligned}
\begin{aligned}[b] & -6 ((((x-2) x+2) y-2) y+1) \left(x \Big(y \left(2 \left(7 c_p+17\right) y x^2-5 \left(8 c_p y+12 y-2 c_p+3\right) x+20 \left(c_p+2\right) y+10\right)\right.\\ & \left.\left.-5\Big)-30 (y-1)\right) m_s^2\right) m_c+3 (x-1) (y-1) \left(22 \left((x ((7 x-41) x+101)-108) x+2 ((x ((x-13) x+38)-44)\right.\right.\\ & \left.x+15) c_p+40\right) x^2 y^6+\left((x ((x+367) x+377)-2040) (-x)+4 (x (2 (28 x-61) (x-1) x+305)-185) c_p\right.\\ & \left.-1370\right) x y^5+\left(3 ((11 x (5 x+33)-864) x+440) x+(2 (((287-45 x) x-1041) x+400) x+100) c_p+470\right) y^4\\ & -\left(((558 x-757) x+5) x+2 (((49 x-421) x+20) x+80) c_p+900\right) y^3+\left((181 x-315) x-2 (62 x+105) c_p x\right.\\ & \left.\left.\left.+70 c_p+620\right) y^2+5 \left(3 x+2 (7 x+1) c_p\right) y-170 y-10 c_p+20\right) s x m_s\right) F(s,x,y)^2-6 (y-1) \left(14 (y ((x ((x\right.\\ & -3) x+3) y-3) y+3)-1) \left(x \left(y \left(2 \left(7 c_p+17\right) y x^2-5 \left(8 c_p y+12 y-2 c_p+3\right) x+20 \left(c_p+2\right) y+10\right)-5\right)\right.\\ & \left.-30 (y-1)\right) m_s^2 m_c^3-7 s x (y ((x ((x-3) x+3) y-3) y+3)-1) \left(10 (y+1) y+(-30 (y+4) y+((22 (7 x-8) y x\right.\\ & -419 x+330) y+346) x y-125) x y+2 (y ((((2 y (34 x+11 ((x-9) x+5) y+75)-107) x-110 y+30) y\\ & \left.+30) x+10 y)-5) c_p+20\right) m_s m_c^2-s (x-1) (y-1) \left(s x^2 y^2 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(50 \left(-(x\right.\right.\right.\\ & \left.-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^3+14 \left((131 x+3) x+4 ((5 x+2) x+5) c_p+75\right) y^2+7 \left((82 x-30) c_p\right.\\ & \left.\left.-35 (x+4)\right) y+385\right)-42 ((((x-2) x+2) y-2) y+1) \left(y \left(4 \left((17 x-30) x+((7 x-20) x+10) c_p+20\right) x y^2\right.\right.\\ & \left.\left.\left.+5 \left(x \left(-x+(8 x-4) c_p-6\right)-6\right) y-10 x \left(c_p-2\right)+30\right)-15\right) m_s^2\right) m_c-21 s^2 (x-1) x (y-1) y \left(\left(4 \left((x ((7 x\right.\right.\right.\\ & \left.-41) x+101)-108) x+2 ((x ((x-13) x+38)-44) x+15) c_p+40\right) x^2 y^5+\left((x ((x+67) x+51)-352)\right.\\ & \left. (-x)+4 (x (((11 x-38) x+36) x+43)-30) c_p-250\right) x y^4+\left(\left((x (35 x+163)-468) x+2 (((39-5 x) x\right.\right.\\ & \left.\left. -189) x+90) c_p+300\right) x+70\right) y^3-\left(\left(16 \left(c_p+6\right) x^2-\left(188 c_p+193\right) x+70 c_p+125\right) x+140\right) y^2\\ & \left. \left.\left.+\left(\left(9 x-2 (18 x+5) c_p+5\right) x+120\right) y+5 \left(2 c_p-1\right) x-50\right) y+10\right) m_s\right) F(s,x,y)+6 (y ((x ((x-3) x+3) y\\ & -3) y+3)-1) (y-1)^2 s m_c \left((x-1) (y-1) \left(\left(4 \left(-(x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^2+14 \left(\left(9 x+4 c_p+3\right)\right.\right.\right.\right.\\ & \left.\left. \left.x+5\right) y+7 \left(4 c_p-5\right) x-70\right) y+35\right) s^2 x^2 y^3+7 \left(10 (y-1) y+\left(y (x ((4 (7 x-8) y x-77 x+82) y+44)\right.\right.\\ & \left.\left.-30 y)+2 (2 y (9 x+2 ((x-9) x+5) y-5)-5) (x y-1) c_p-25\right) x y+10\right) s x m_c m_s y+14 \left(\left(-4 \left((17 x-30) x\right.\right.\right.\\ & \left.\left.\left.\left.\left.+((7 x-20) x+10) c_p+20\right) x y^2+5 \left(\left(x+(4-8 x) c_p+6\right) x+6\right) y+10 \left(c_p-2\right) x-30\right) y+15\right) m_c^2 m_s^2\right)\right)\\ & +\frac{c_3}{23592960 \pi ^5 (y-1)^3} (x-1) x^2 \left(c_p+1\right) F(s,x,y)^2 \Big(8 s (y-1) y ((393 x-200) y+73) F(s,x,y)+((477 x\\ & -230) y+43) F(s,x,y)^2+24 s^2 (y-1)^2 y^2 ((64 x-30) y+15)\Big)\Bigg\},\\ \rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Big\{-\frac{c_1 x }{3072 \pi ^3 (y-1)^3}m_c F(s,x,y) \left(3 s (y-1) \left(y c_p \left(x \left(2 y \left(22 (x-1) x y^2-4 (x-1) (3 x+5)y\right.\right.\right.\right.\right.\\ & \left.\left.\left.\left.+13 x-9\right)-7\right)-2 y\right)+c_p+y (x (y (x (y (66 (x-1) y-41 x-23)+42)+69 y-28)-15)-21 y+19)-2\right)\\ & F(s,x,y)+\Big(x \left(y \left(2 (x-1) c_p (y (8 x y-2 x-7)+1)+x (y (24 (x-1) y-9 x-13)+7)+25 y-12\right)+2\right)-y\\ & +1\Big) F(s,x,y)^2+6 s^2 (y-1)^2 y \left(y \left(x \left(2 c_p (x y-1) (2 (x-1) (2 y-1) y+1)+y (x (y (12 (x-1) y-7 x-3)+6)\right.\right.\right.\\ & \left.\left.\left.\left.+11 y-2)-4\right)-5 y+5\right)-1\right)\right)-\frac{c_2 x^2}{2048 \pi ^3 (y-1)^3} m_c F(s,x,y) \left(3 s (y-1) y \left(x y \left(44 (x-1) y^2+(26-39 x) y\right.\right.\right.\\ & \left.\left.+11\right)+18 (y-1) y+2\right) F(s,x,y)+(x y (y (16 (x-1) y-13 x+6)+4)+10 (y-1) y+3) F(s,x,y)^2+6 s^2\\& \left.(y-1)^2 y^3 \left(8 (x-1) x y^2+(x (6-7 x)+2) y+x-2\right)\right)\Bigg\},\\ \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1}{9216 \pi ^3 (y-1)^2} \left(c_p \left(12 s x (y-1) y F(s,x,y) \left(6 m_c^2 (2 y (((x-4) x+2) y+2 x-1)-1)\right.\right.\right.\\ & \left.+s (x-1) (y-1) y^2 \left(x \left(50 \left(x^2-2\right) y^2+8 (x+7) y+11\right)-4 y+3\right)\right)+3 F(s,x,y)^2 \left(12 x y m_c^2 (((x-4) x+2) y\right.\\ & \left.+x)+s (x-1) (y-1) \left(2 x y \left(y \left(7 x \left(y \left(20 \left(x^2-2\right) y+7 x+24\right)-1\right)-22 y+11\right)+3\right)+4 (y-1) y+1\right)\right)\\ \end{aligned}
\begin{aligned}[b] & +(x-1) \left(x \left(y \left(x \left(8 y \left(10 \left(x^2-2\right) y+9 x+8\right)-21\right)-16 y+4\right)+3\right)+2 (y-1)\right) F(s,x,y)^3+24 s^3 (x-1) x^2\\ & \left. (y-1)^3 y^4 \left(2 y \left(\left(x^2-2\right) y+1\right)+1\right)\right)+x \left(-6 s (y-1) y F(s,x,y) \left(6 m_c^2 (y (x+8 y-12)+5)-s (x-1) (y-1)\right.\right.\\& \left. y (y (x (2 y (25 (5 (x-2) x+6) y+119 x-183)+139)-30 y+28)-11)\right)+3 F(s,x,y)^2 \left(s (x-1) (y-1) y (y \right.\\ & \left.(x (14 y (x (50 (x-2) y+51)+60 y-72)+313)-100 y+82)-21)-6 m_c^2 (y (3 x+4 y-8)+1)\right)+(x-1)\\ & (y (x (8 y (5 (5 (x-2) x+6) y+32 x-39)+51)-40 y+30)-1) F(s,x,y)^3+6 s^3 (x-1) (y-1)^3 y^3 \Big(y \Big(x \Big(4 (5\\ & \left.\left. (x-2) x+6) y^2+6 (3 x-5) y+13\Big)-2 y+2\Big)-1\Big)\right)\right)-\frac{c_2}{8192 \pi ^3 (y-1)^3} x \left(2 \left(6 s (y-1)^2 y F(s,x,y) \left(m_c^2 (2 x y\right.\right.\right.\\ & \left. (3-4 x y)-2)+s x y^2 (x (x y (5 y (10 (x-1) y-10 x+3)+36)+7 (5 y-4) y-8)-9 y+9)\right)+3 (y-1)\\ & F(s,x,y)^2 \left(s x y (y (x (5 x y (2 y (14 (x-1) y-14 x+5)+19)+90 (y-1) y-7)-10 y+14)-2)-2 m_c^2 (x y-1)\right.\\ & \left. (2 x y+1)\right)+x (y (x (y (x (20 y (2 x (y-1)-2 y+1)+23)+20 y-34)+8)+8 y-2)-3) F(s,x,y)^3+6 s^3\\ & \left. (x-1) x (y-1)^4 y^4 (x y (4 x y-3)+1)\right)+c_p \left(3 s (y-1) \left(y \left(x \left(2 y \Big(22 (x-1) x y^2-6 (x-1) (2 x+5) y+23 x\right.\right.\right.\right.\\ & \left.\left.\left.-24\Big)+3\right)-4 y+4\right)-1\right) F(s,x,y)+2 (x-1) (x y (y (8 x y-2 x-13)+7)+y-1) F(s,x,y)^2+12 s^2 x (y-1)^2\\ & \left.\left. y^2 \left(y \left(4 (x-1) x y^2-2 (x-1) (x+3) y+5 x-6\right)+1\right)\right) F(s,x,y)\right)\Bigg\},\\ \rho^{\langle\bar{s}s\bar{s}s\rangle}_{2,8;A(S)}(s) =& \int^{z_{max}}_{z_{min}}{\rm d}z\frac{c_1 c_p}{768 \pi } m_s^2 G(s,z) \left(G(s,z)+s (1-2 z)^2\right),\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{2,8;A(S)}(s) =& \int^{z_{\max}}_{z_{\min}}{\rm d}z\Bigg\{-\frac{c_1}{2304 \pi } \left(m_c m_s \left(s \left(z^2 \left(-\left(6 c_p+23\right)\right)+3 \left(c_p-2\right)+17 z\right)-11 G(s,z)\right)+m_s^2 \left(\left(-8 z^2 \left(3 c_p+5\right)\right.\right.\right.\\ & \left.\left.+2 z \left(7 c_p+15\right)+c_p-1\right) G(s,z)-s (z-1) z \left(6 (11 z-8) z c_p+c_p+2 (50 z-41) z+21\right)\right)+6 m_c^2 (2 G(s,z)\\ & \left.+2 s (z-1) z+s)\right)+\frac{c_2 c_p}{2048 \pi } m_s^2 \left(2 G(s,z)+s (1-2 z)^2\right)\Bigg\}+\int^{1}_{0}{\rm d}z\Bigg\{\frac{c_1 s^2 (z-1) z}{2304 \pi } m_s \left((z-1) z m_s \left((2 (6 z-5) z\right.\right.\\ & \left.\left.+1) c_p+30 z^2-28 z+11\right)+(7 (z-1) z+3) m_c\right)\Bigg\},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{2,8;A(S)}(s) =& \int^{x_{max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^2}{18432 \pi ^3 (x-1)^2 (y-1)^3} m_c \left(2 F(s,x,y) \left(m_c^2 (x y-1)^2 (y (((x-3) x+3) y+x-3)+1)\right.\right.\\ & \left. (8 x y+1)+3 s (x-1) (y-1) y (x y (y (x (y (22 (x-1) y-17 x-19)+29)+41 y-28)-4)-13 (y-1) y-2)\right)\\ & +3 (x-1) (x y-1) (x (8 y-5) y ((x-1) y-1)+11 (y-1) y+3) F(s,x,y)^2+2 s (y-1) y \left(m_c^2 (11 x y-5) (x y-1)\right.\\ & (2 x y-1) (y (((x-3) x+3) y+x-3)+1)+3 s (x-1) (y-1) y^2 \left(y \left(x \Big(x \left(4 (x-1) y^2-3 (x+1) y+5\right)+7 y\right.\right.\\ & \left.\left.\left.\left.-6\Big)-1\right)+1\right)\right)\right)\Bigg\}-\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s^2 x^2 y^2}{9216 \pi ^3 (x-1)^2 (y-1)} m_c^3 (x y-1) (x y (4 x y-3)+1) (y (((x-3) x+3) y\\ & +x-3)+1),\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x}{18432 \pi ^3 (x-1)^3 (y-1)^3} m_c^2 \left((x-1) \left(c_p (x y-1) (y (((x-3) x+3) y+x-3)+1) (x (y (8\right.\right.\\& ((x-4) x+2) y+15 x-4)-3)-2 y+2)-40 x^3 ((x-3) x+3) ((x-2) x+2) y^6+8 \Big(\left(x^3+14 x-36\right) x\\ & +45\Big) x^2 y^5+3 (x (x (3 (x-21) x+131)-166)-24) x y^4+(x (((41 x-95) x+224) x+162)-24) y^3+((x (4 x\\ & \left.-45)-109) x+48) y^2+3 (x (x+7)-12) y-x+12\right) F(s,x,y)+4 m_c^2 (x y-1) (y (((x-3) x+3) y+x-3)\\ & +1) \left(((x-2) x+2) x y^2-(x+1) y+1\right)+s (x-1) (y-1) \left(c_p (2 y (x (y (11 ((x-4) x+2) y+25 x-11)-3)\right.\\ & -2 y+2)-1) (x y-1) (y (((x-3) x+3) y+x-3)+1)+y \left(-140 x^3 ((x-3) x+3) ((x-2) x+2) y^6+2 ((x\right.\\ & ((11 x+35) x+89)-366) x+570) x^2 y^5-x (((x (33 x+427)-1041) x+1686) x+204) y^4+(x (((161 x\\ \end{aligned}
\begin{aligned}[b] & \left.\left.\left.-361) x+1078) x+402)-24) y^3+((x (20 x-379)-277) x+48) y^2+17 (3 x+5) x y-3 x-48 y+24\right)-6\right)\right)\Bigg\}\\ &+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Big\{\frac{c_1 s x}{18432 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \left(2 m_c^2 \left(y \left(x \left(4 ((x-2) x\right.\right.\right.\right.\\ & \left.\left.\left.+2) y^2-6 y+3\right)-2 y+2\right)-1\right)-s (x-1) x (y-1) y^2 \left(y \left(-2 c_p (2 ((x-4) x+2) y+5 x-3)+50 ((x-2) x\right.\right.\\ & \left.\left.\left.+2) x y^2-2 (x+15) (2 x+1) y+29 x+28\right)-11\right)\right)\Big\},\\ \rho^{\langle g_sG^2\rangle^2}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^3 y^3}{371589120 \pi ^5 (x-1)^2 (y-1)^2} m_c^4 \Big(c_p (x (x y (8 y (5 x (20 x y+21)-28 (5 y+4))+483)\\ & +140 (4 y-1) y-105)-70 (y-1))+x (x y (8 y (7 (38 x-21)-5 ((x-14) x+42) y)+231)+350 (4 y-3) y\\ & +35)\Big)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x^3 y^3 }{53084160 \pi ^5 (x-1)^2 (y-1)}m_c^4 \left(c_p \left(2 x y \left(x y \left(y \left(40 \left(5 x^2-7\right) y+115 x-96\right)+143\right)\right.\right.\right.\\ & \left.\left.+55 (2 y-1) y-15\right)-20 (y-1) y-5\right)+x y \left(y \Big(x \left(-20 ((x-14) x+42) y^2+6 (141 x-32) y-29\right)+500 y\right.\\ & \left.\left.-410\Big)+105\right)\right)\Bigg\},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{2,8;A(S)}(s) =& \int^{x_{max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x}{36864 \pi ^3 (x-1)^3 (y-1)^3} m_c \left(3 (x-1) \left(8 (2 x-3) (x-1) x^2 y^5 \left(2 (x-1) c_p+x+1\right)+y^3\right.\right.\\ & \left(2 (x (2 x-27)-2) (x-1)^2 c_p+\left(\left(22 x^2-30 x-111\right) x+94\right) x+13\right)-x y^4 \left(10 (2 (x-1) x-5) (x-1)^2 c_p\right.\\ & \left.+((x (x+58)-149) x+50) x+37\right)+y^2 \left(2 (8 x+3) (x-1)^2 c_p+((92-13 x) x-33) x-28\right)-y \Big(2 (x-1)^2 c_p\\ & \left.+(15 x+16) x-19\Big)+7 x-4\right) F(s,x,y)+m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \left(x \left(y \left(2 (x-1) c_p (y (x (8 y\right.\right.\right.\\ & \left.\left.\left.-2)-7)+1)+x (y (24 (x-1) y-9 x-13)+7)+25 y-12\right)+2\right)-y+1\right)+3 s (x-1) (y-1) \Big(22 (2 x-3)\\ & (x-1) x^2 y^6 \left(2 (x-1) c_p+x+1\right)-x y^5 \left(4 (x (17 x-4)-17) (x-2) (x-1) c_p+((x (19 x+98)-329) x\right.\\ & \left.+118) x+89\right)+y^4 \left(2 (x (x ((7 x-89) x+177)-87)-5) c_p+((x (67 x-104)-231) x+246) x+5\right)+y^3\\ & \left((((45 x-127) x+47) x+16) c_p+((205-36 x) x-132) x-14\right)+y^2 \left((4 x (3 x+4)-7) c_p+5 x (2-5 x)-2\right)\\ & \left.-y \left(8 x c_p+c_p+x-9\right)+c_p-2\Big)\right)-\frac{c_2 x^2 y}{24576 \pi ^3 (x-1)^2 (y-1)^2} m_c^3 (x y-1) \Big(y (x (y (8 ((x-3) x+3) y+6 x\\& -15)+3)-9 y+9)-2\Big)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{c_1 s x}{36864 \pi ^3 (x-1)^3 (y-1)^2} m_c \left(m_c^2 (y (y (x ((x-3) x+3) y-3)\right.\\ & +3)-1) \left(y c_p \left(x \left(2 y \left(22 (x-1) x y^2-4 (x-1) (3 x+5) y+13 x-9\right)-7\right)-2 y\right)+c_p+y (x (y (x (y (66 (x-1) y\right.\\ & \left.-41 x-23)+42)+69 y-28)-15)-21 y+19)-2\right)+3 s (x-1) (y-1) y \left(y \left(4 (2 x-3) (x-1) x^2 y^5 \Big(2 (x\right.\right.\\& -1) c_p+x+1\Big)-x y^4 \left(4 (x (3 x-1)-3) (x-2) (x-1) c_p+(3 x (x+8)-11) (x-2) x+15\right)+y^3 \left(2 (x ((x-15)\right.\\ & \left.x+33)-18) x c_p+(x (11 x-6)-48) (x-1) x-3\right)+y^2 \left(x \left(2 (2 (2 x-7) x+7) c_p+x (37-6 x)-33\right)+6\right)\\ & \left.\left.\left. y \left((4 x+2) c_p-5 x+9\right)-2 x \left(c_p+1\right)-8 y+5\right)-1\right)\right)-\frac{c_2 s x^2 y^2}{24576 \pi ^3 (x-1)^2 (y-1)} m_c^3 \Big(x y (y (x (y (22 ((x\\ & -3) x+3) y-4 x+17)-2)-83 y+60)-17)+23 (y-1) y+7\Big)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{2,8;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1}{110592 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \left(8 x^2 y^5 \left((((x (9 x-19)-29) x+30) x+54) c_p+(((32 x\right.\right.\\ & \left.-135) x+133) x+48) x-105\right)+40 ((x-3) x+3) x^3 y^6 \left(2 \left(x^2-2\right) c_p+5 (x-2) x+6\right)+x y^4 \left((x ((15 x+91)\right.\\ & \left.x+143)-1098) x c_p+198 c_p+3 ((x (17 x+159)-659) x+630) x-24\right)+y^3 \left((x (x (x (61-41 x)+790)\right.\\ & \left.-210)-6) c_p+(((1423-255 x) x-1728) x+222) x\right)+y^2 \left(((91-x (36 x+343)) x+12) c_p+((609-274 x)\right.\\ \end{aligned}
\begin{aligned}[b]& \left.\left.x-263) x\right)+y \left((x (57 x+5)-8) c_p+21 x (7-5 x)\right)-3 x c_p+2 c_p-17 x\right)+\frac{c_2 x y}{49152 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \Big(y^2 \\ & \left(x^3 \left(-\left(c_p+3\right)\right)+x^2 \left(94-33 c_p\right)+3 x \left(c_p+7\right)+4 c_p\right)+4 x y^4 \left(2 ((x ((x-4) x+6)-6) x+2) c_p+(((2 x-11)\right. \\ & \left.x+21) x+16) x+10\right)+y^3 \left((x (x ((5 x-14) x+54)-18)-2) c_p+(3 x (x+1) (8 x-31)-70) x\right)+y \left((2 x (5 x\right.\\ & \left.+2)-3) c_p+7 x (1-3 x)+6\right)-2 x c_p+c_p-40 x^2 ((x-2) ((x-2) x+2) x+2) y^5-2 x-6\Big)\Bigg\}\\ & +\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1}{110592 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(c_p \left(-12 x y m_c^2 (x y-1) (((x-4) x+2) y+x) (y (((x-3) x+3) y\right.\right.\\ & +x-3)+1)-s (x-1) (y-1) \left(y \left(2 (((7 x (7 x+3)-799) x+570) x+774) x^2 y^5+280 \left(x^2-2\right) ((x-3) x+3)\right.\right.\\ & x^3 y^6+2 ((x ((29 x+236) x+233)-1749) x+216) x y^4-2 (((x (65 x+116)-1324) x+321) x+6) y^3\\ & \left.\left.\left.+((311-x (61 x+885)) x+24) y^2+(2 x (79 x-2)-19) y-42 x+7\right)-1\right)\right)+x \left(6 m_c^2 (x y-1) (y (3 x+4 y-8)\right.\\ & +1) (y (((x-3) x+3) y+x-3)+1)-s (x-1) (y-1) y \left(y \left(140 ((x-3) x+3) (5 (x-2) x+6) x^2 y^5+2 (x ((21\right.\right.\\ & (17 x-75) x+1483) x+738)-1410) x y^4+(((x (313 x+1243)-5793) x+6078) x+12) y^3+(((3857\\& \left.\left.\left.\left.-739 x) x-5334) x+174) y^2+(3 (721-268 x) x-347) y-331 x+251\right)-69\right)\right)\right)\\ & -\frac{c_2 x y}{98304 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(s (x-1) (y-1) \left(y c_p \left(-44 x ((x ((x-4) x+6)-6) x+2) y^4+4 ((((23-8 x) x\right.\right.\right.\\ & \left.-82) x+31) x+2) y^3-2 (((x-112) x+28) x+8) y^2+(14-x (67 x+14)) y+14 x-6\right)+c_p+2 \left(x y \left(y \Big(140\right.\right.\\ & ((x-2) ((x-2) x+2) x+2) x y^4-2 ((((11 x-58) x+108) x+158) x+50) y^3+((5 x (23-8 x)+519) x\\ & \left.\left.\left.+182) y^2+(2 (x-182) x-135) y+73 x+57\Big)-5\right)-18 (y-1) y-6\right)\right)-4 m_c^2 \left(2 ((x-2) ((x-2) x+2) x+2)\right.\\ & \left.\left.x y^4+((x-4) x (x+1)-2) y^3+(x (6-(x-3) x)+4) y^2-(5 x+3) y+x+1\right)\right)\Bigg\},\\ \end{aligned}
(24) where the coefficient
c_p=1 for currentJ_{2,\mu\nu}^{A(S)} andc_p=-1 for currentJ_{8,\mu\nu}^{A(S)} . The spectral functions for the(1,1\{0,1\}) structure are given as\begin{aligned}[b] \rho^{pert}_{3,9;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{{\rm i} x^2}{51609600 \pi ^5 (y-1)^5} F(s,x,y)^3 c_1 \left((x-1) x \left(-40 \left((27 x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x^2\right.\right.\\ & y^3-56 x \left(7 (12 x-7) x+((3 x-4) x+10) c_p+25\right) y^2+7 \left(x (140-57 x)+(x (50-99 x)+10) c_p\right) y\\ & \left.+35 \left(x+(3 x-2) c_p\right)\right) F(s,x,y)^3+42 x \left(\left(-20 c_p (y-1)-110 (y-1)-75 x+((y (443 x-8 ((27 x-88) x\right.\right.\\ & +40) y-1026)-120) x+470 y+140) x y+2 (y ((50-y (41 x+8 ((x-9) x+5) y+38)) x+40 y-20)-15) x\\ & \left. c_p\right) m_c m_s-s (x-1) (y-1) y \left((100 (5 y-4) y+(2 (909 x+10 (x (27 x-14)-42) y-364) y+127) x y+95) x\right.\\ & \left.\left.+2 \left(x \left(y \left(\left(\left(3 x+40 \left(5 x^2-7\right) y+16\right) y+144\right) x+110 y-80\right)-15\right)-10 (y-1)\right) c_p\right)\right) F(s,x,y)^2+30 (y-1)\\& \left((x-1) (y-1) \left(50 \left((27 x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^3-14 \left(-303 x^2+93 x+4 (2 x-5) (3 x+1) c_p-75\right)\right.\right.\\ & \left.y^2+7 \left(37 x+20 (3 x-2) c_p-130\right) y+315\right) \left(-s^2\right) x^2 y^2+14 \left(x \left(y \left(-75 x+2 \left((37 x-50) x+((7 x-20) x\right.\right.\right.\right.\\ & \left.\left.\left.\left.+10) c_p+20\right) y+5 (6 x-4) c_p+70\right)-5\right)-30 (y-1)\right) m_c^2 m_s^2-7 s x \left(\left((-10 (115 y+12) y+((-719 x+22 ((27 \right.\right.\\ & x-88) x+40) y+2190) y+26) x y+95) x+350 y+4 (10 (y-1)+(((y (59 x+11 ((x-9) x+5) y+50)-71) x\\ & \left.\left.\left.-55 y+35) y+15) x) c_p-270\right) y+60\right) m_c m_s\right) F(s,x,y)-120 s (y-1)^2 \left((x-1) (y-1) \left(70 (y-1) y+\left(4 \Big((27 x\right.\right.\right.\\ & \left.\left.-14) x+4 \left(5 x^2-7\right) c_p-42\Big) y^2-14 \left(-23 x+4 (x-2) c_p+5\right) y+21\right) x y+35\right) s^2 x^2 y^3+7 \left(50 (y-1) y\right.\\ & +\Big(\left((y (-97 x+4 ((27 x-88) x+40) y+342)-16) x-190 y+8 (7 x+((x-9) x+5) y-5) (x y-1) c_p+20\right) y\\ & \left.+5\Big) x y+20\right) s x m_c m_s y+14 \left(30 (y-1) y+\left(-4 \left(\left(7 c_p+37\right) x^2-10 \left(2 c_p+5\right) x+10 \left(c_p+2\right)\right) y^2-5 (3 x-2)\right.\right.\\ & \left.\left.\left.\left. \left(4 c_p-3\right) y+10\right) x y+15\right) m_c^2 m_s^2\right)\right), \end{aligned}
\begin{aligned}[b] \rho^{\langle\bar{s}s\rangle}_{3,9;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^2}{3072 \pi ^3 (y-1)^4} m_c F(s,x,y)^2 \left(4 s (y-1) (y (x (y (x (y (22 (x-3) (2 x-1) y-34 x+153)\right.\\ & -12)-75 y-4)+9)+19 y-15)+3) F(s,x,y)+(x y-1) (x (y (8 (x-3) (2 x-1) y-8 x+13)-5)-5 y+5)\\ & \left. F(s,x,y)^2+12 s^2 (y-1)^2 y (y (x y (x (y (4 (x-3) (2 x-1) y-4 x+25)-4)-13 y+2)+x+3 (y-1))+1)\right),\\ \rho^{\langle m_s\bar{s}s\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x}{6144 \pi ^3 (y-1)^3} F(s,x,y) \left(12 s (y-1) F(s,x,y) \left(2 m_c^2 (y (x (2 y (2 ((5 x-6) x+2) y-2 x+1)\right.\right.\\ & \left.-1)-2 y+2)-1)+s (x-1) x (y-1) y^2 (y (x (2 y (25 (x (3 x-2)-2) y+130 x-21)+11)+30 y-26)+9)\right)\\ & +4 F(s,x,y)^2 \left(4 m_c^2 (y (x (((5 x-6) x+2) y-4 x+3)-1)+1)+s (x-1) x (y-1) y (x y (2 y (70 (x (3 x-2)-2) y\right.\\ &\left.+385 x-88)+35)+20 (5 y-4) y+19)\right)+(x-1) c_p \left(8 s (y-1) y (x (y (11 ((x-4) x+2) y+30 x-16)-3)\right.\\ & -2 y+2) F(s,x,y)+(x (y (8 ((x-4) x+2) y+21 x-10)-3)-2 y+2) F(s,x,y)^2+48 s^2 x (y-1)^2 y^3 (((x-4)\\ & \left. x+2) y+3 x-2)\right) F(s,x,y)+(x-1) x (y (x (8 y (5 (x (3 x-2)-2) y+35 x-13)+13)+40 y-28)-1)\\ & \left. F(s,x,y)^3+24 s^3 (x-1) x (y-1)^3 y^3 \left(y \left(4 (x (3 x-2)-2) x y^2+2 ((10 x-1) x+1) y+x-2\right)+1\right)\right), \end{aligned}
\begin{aligned} \rho^{\langle g_sG^2\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{x^2 c_1}{30965760 \pi ^5 (x-1)^3 (y-1)^5} m_c \left((x-1) x \left(\left(y \left(40 (x (27 x-14)-42) x y^2+56 (7 (12 x-7)\right.\right.\right.\right.\\ & \left.\left. x+25) y+399 x-980\right)-35\right) (x y-1) ((x+((x-3) x+3) y-3) y+1) x m_c+21 \left(8 ((x (3 (9 x-47) x+281)\right.\\ & -268) x+80) x^2 y^5+(x (((1743-451 x) x-2629) x+2836)-1050) x y^4+((((121-15 x) x-900) x+120) x\\ & \left.+370) y^3+(x ((38 x-175) x+695)-660) y^2+5 (3 (9 x-23) x+74) y+45 x-80\right) m_s+c_p \Big((x y-1)\\ & ((x+((x-3) x+3) y-3) y+1) \Big(x \left(70 (8 y-5) y+\left(8 y \left(7 (3 x-4)+20 \left(5 x^2-7\right) y\right)+693\right) x y-105\right)-70\\& (y-1)\Big) m_c+42 \Big(8 ((x ((x-13) x+38)-44) x+15) x^2 y^5+(x (((57 x-191) x+173) x+188)-140) x y^4\\ & +((((103-15 x) x-420) x+180) x+30) y^3-(((16 x-185) x+25) x+60) y^2-35 x (x+1) y+40 y+15 x\\ & \left.-10\Big) m_s\Big)\right) F(s,x,y)^3+21 \left((y ((x ((x-3) x+3) y-3) y+3)-1) x \left(110 (y-1)+(-10 (47 y+14) y+((-443 x\right.\right.\\ & +8 ((27 x-88) x+40) y+1026) y+120) x y+75) x+2 (10 (y-1)+(((y (41 x+8 ((x-9) x+5) y+38)-50)\\ & \left.x-40 y+20) y+15) x) c_p\right) m_s m_c^2+(x-1) (y-1) \left(s x y (y ((x ((x-3) x+3) y-3) y+3)-1) \left((100 (5 y-4) y\right.\right.\\ & +(2 (909 x+10 (x (27 x-14)-42) y-364) y+127) x y+95) x+2 \left(x \left(y \left(\left(\left(3 x+40 \left(5 x^2-7\right) y+16\right) y+144\right)\right.\right.\right. \\ & \left.\left.\left.\left.x+110 y-80\right)-15\right)-10 (y-1)\right) c_p\right)-6 ((((x-2) x+2) y-2) y+1) \left(x \left(y \left(-75 x+2 \left((37 x-50) x+((7 x\right.\right.\right.\right.\\ & \left.\left.\left.\left.\left.-20) x+10) c_p+20\right) y+5 (6 x-4) c_p+70\right)-5\right)-30 (y-1)\right) m_s^2\right) m_c+3 (x-1) (y-1) \Big(22 \left((x (3 (9 x-47)\right.\\ & \left.x+281)-268) x+2 ((x ((x-13) x+38)-44) x+15) c_p+80\right) x^2 y^6+\left((((2833-741 x) x-4217) x+5860)\right.\\ & \left.x+4 (x (((81 x-278) x+272) x+230)-185) c_p-2490\right) x y^5+\left((((109-15 x) x-2952) x+1200) x+4\right.\\ & \left.((((121-15 x) x-573) x+270) x+30) c_p+810\right) y^4-\left((x (18 x-877)-235) x+4 \Big(\left(22 x^2-278 x+85\right) x\right.\\ & \left.+60\Big) c_p+1460\right) y^3+\left((41 x-535) x-16 (x (14 x+5)-10) c_p+1160\right) y^2+5 \left(27 x+4 (3 x-2) c_p-86\right) y\\& \left.+60\Big) s x m_s\right) F(s,x,y)^2-6 (y-1) \left(14 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(x \left(y \left(-75 x+2 \left((37 x-50) x\right.\right.\right.\right.\right.\\ & \left.\left.\left.\left.+((7 x-20) x+10) c_p+20\right) y+5 (6 x-4) c_p+70\right)-5\right)-30 (y-1)\right) m_s^2 m_c^3-7 s x (y ((x ((x-3) x+3) y-3)\\ & y+3)-1) \left(\left((-10 (115 y+12) y+((-719 x+22 ((27 x-88) x+40) y+2190) y+26) x y+95) x+350 y+4\right.\right.\\ & \left.\left.(10 (y-1)+(((y (59 x+11 ((x-9) x+5) y+50)-71) x-55 y+35) y+15) x) c_p-270\right) y+60\right) m_s m_c^2\\ & -s (x-1) (y-1) \left(s x^2 y^2 (y ((x ((x-3) x+3) y-3) y+3)-1) \left(50 \left((27 x-14) x+4 \left(5 x^2-7\right) c_p-42\right) x y^3\right.\right.\\ & \left.-14 \left(-303 x^2+93 x+4 (2 x-5) (3 x+1) c_p-75\right) y^2+7 \left(37 x+20 (3 x-2) c_p-130\right) y+315\right)-42 ((((x-2)\\ & x+2) y-2) y+1) \left(4 \left(\left(7 c_p+37\right) x^2-10 \left(2 c_p+5\right) x+10 \left(c_p+2\right)\right) x y^3+5 \left(x (3 x-2) \left(4 c_p-3\right)-6\right) y^2\right.\\ \end{aligned}
\begin{aligned}[b]& \left.\left.-10 (x-3) y-15\right) m_s^2\right) m_c-21 s^2 (x-1) x (y-1) y \left(\left(4 \left((x (3 (9 x-47) x+281)-268) x+2 ((x ((x-13) x\right.\right.\right.\\& \left.+38)-44) x+15) c_p+80\right) x^2 y^5+\Big((((373-101 x) x-531) x+892) x+8 (x (((8 x-29) x+33) x+14)-15)\\& c_p-410\Big) x y^4+\left(\left(((23-5 x) x-588) x+24 ((2 x-17) x+10) c_p+320\right) x+110\right) y^3-\left(\left((6 x-223) x+8 ((2 x\right.\right.\\ & \left.\left.\left.\left.\left.-31) x+20) c_p+95\right) x+220\right) y^2-x \left(11 x+8 (7 x-5) c_p+35\right) y+210 y+15 x-100\right) y+20\right) m_s\right) F(s,x,y)\\ & -6 s (y-1)^2 (y ((x ((x-3) x+3) y-3) y+3)-1) m_c \left((x-1) (y-1) \left(70 (y-1) y+\left(4 \left((27 x-14) x+4 \left(5 x^2-7\right)\right.\right.\right.\right.\\ & \left.\left.\left. c_p-42\right) y^2-14 \left(-23 x+4 (x-2) c_p+5\right) y+21\right) x y+35\right) \left(-s^2\right) x^2 y^3-7 s x \left(50 (y-1) y+\left(\left((y (-97 x+4 ((27\right.\right.\right.\\ & \left.\left.\left. x-88) x+40) y+342)-16) x-190 y+8 (7 x+((x-9) x+5) y-5) (x y-1) c_p+20\right) y+5\right) x y+20\right) m_c m_s\\ & y+14 \left(4 \left((37 x-50) x+((7 x-20) x+10) c_p+20\right) x y^3+5 \left(x (3 x-2) \left(4 c_p-3\right)-6\right) y^2-10 (x-3) y-15\right)\\ & m_c^2 m_s^2\Big)\Big)\Bigg\}, \end{aligned}
\begin{aligned}[b]\rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x}{3072 \pi ^3 (y-1)^3} m_c F(s,x,y) \left(3 s (y-1) \left(y \left(4 (x-1) (y-1) c_p (x y (11 x y-10)+1)\right.\right.\right.\\ & \left.\left.+x (x y (y (22 ((6 x-25) x+9) y-73 x+517)-51)-7 (29 y+2) y+30)+41 y-33\right)+6\right) F(s,x,y)\\& +\left(2 (x-1) (y-1) c_p (x y (8 x y-7)+1)+x y (x (y (8 ((6 x-25) x+9) y-65 x+223)+11)-75 y-26)+12 x\right.\\ & \left.+9 y-9\right) F(s,x,y)^2+6 s^2 (y-1)^2 y \left(x y \left(y \left(8 (x-1) (y-1) c_p (x y-1)+x (y (4 ((6 x-25) x+9) y-7 x+89)\right.\right.\right.\\ & -17)-37 y+6\Big)+4\Big)+7 (y-1) y+2\Big)\Big)-\frac{c_2 x^2}{2048 \pi ^3 (x-1) (y-1)^4} m_c F(s,x,y) \left(3 s (y-1) \left(y \left(44 ((x-3) x+1)\right.\right.\right.\\ & ((x-1) x+1) x y^4+(x (((197-56 x) x-47) x+12)-18) y^3+((15-22 x (x+3)) x+32) y^2+3 (4 x+5) x y\\ & \left.\left.-9 x-36 y+18\right)-3\right) F(s,x,y)+\left(16 ((x-3) x+1) ((x-1) x+1) x y^4+(x (((103-32 x) x-53) x+24)\right.\\ & \left.-10) y^3-(x ((7 x+11) x+14)-16) y^2+9 (x-1) y-3 x+3\right) F(s,x,y)^2+6 s^2 (y-1)^2 y \left(y \left(\left(x \left(-8 x^3+29 x^2\right.\right.\right.\right.\\ & \left.\left.\left.+x-4\right)-2\right) y^3+8 ((x-3) x+1) ((x-1) x+1) x y^4+((9-4 x (x+4)) x+4) y^2+(x (4 x-1)-6) y-x+4\right)\\& -1\Big)\Big)\Bigg\}, \end{aligned}
\begin{aligned}[b] \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1}{9216 \pi ^3 (y-1)^2} \left(c_p \left(24 s x (y-1) y^2 F(s,x,y) \left(6 m_c^2 (((x-4) x+2) y+3 x-2)+s\right.\right.\right.\\& \left.(x-1) (y-1) y \left(x \left(25 \left(x^2-2\right) y^2+2 (25-9 x) y-3\right)-2 y+2\right)\right)+6 y F(s,x,y)^2 \left(6 x m_c^2 (((x-4) x+2) y+3 x\right.\\ & \left.-2)+s (x-1) (y-1) \left(x \left(y \left(x \left(7 y \left(20 \left(x^2-2\right) y-9 x+40\right)-36\right)-22 y+16\right)+3\right)+2 (y-1)\right)\right)+(x-1) \\ & \left(x \left(y \left(x \left(8 y \left(10 \left(x^2-2\right) y-3 x+20\right)-27\right)-16 y+10\right)+3\right)+2 (y-1)\right) F(s,x,y)^3+48 s^3 (x-1) x^2 (y-1)^3\\ & \left. y^5 \left(\left(x^2-2\right) y-x+2\right)\right)-x \left(6 s (y-1) y F(s,x,y) \left(6 m_c^2 \left(8 (1-2 x)^2 y^2-3 x y-1\right)+s (x-1) (y-1) y \left(y \left(x \Big(50\right.\right.\right.\right.\\ & \left.\left.\left.(5 x (3 x-2)-6) y^2+(954 x-66) y+29\Big)+30 y-26\right)+9\right)\right)+3 F(s,x,y)^2 \left(6 m_c^2 \Big(y \left(4 (1-2 x)^2 y-9 x+4\right)\right.\\& \left.+1\Big)+s (x-1) (y-1) y (x y (2 y (70 (5 x (3 x-2)-6) y+1401 x-164)+83)+20 (5 y-4) y+19)\right)+(x-1)\\ & (y (x (8 y (5 (5 x (3 x-2)-6) y+126 x-29)+21)+40 y-28)-1) F(s,x,y)^3+6 s^3 (x-1) (y-1)^3 y^3 (x y (2 y\\& \left.\left.(2 (5 x (3 x-2)-6) y+37 x-1)+3)+2 (y-1) y+1)\right)\right)+\frac{c_2 x}{4096 \pi ^3 (x-1) (y-1)^3} \left(6 s (y-1) F(s,x,y)\right.\\ & \left(2 m_c^2 (y (x y (x (y (4 (4 x-3) (x-1) y-12 x+25)-4)-13 y+2)+x+3 (y-1))+1)+s (x-1) x (y-1) y^2 (y (x\right.\\ & \left.(x y (y (50 (2 x-3) (x+1) y+62 x+65)-148)+5 (33 y-4) y-11)-37 y+33)-9)\right)+3 (x-1) F(s,x,y)^2\\ & \left(2 m_c^2 (y (2 (4 x-3) x y-2 x+1)-1) (x y-1)+s x (y-1) y (y (x (y (x (2 y (70 (2 x-3) (x+1) y+108 x+55)-415)\right.\\ \end{aligned}
\begin{aligned}[b]& \left.+470 y-26)-38)-108 y+92)-21)\right)-(x-1)^2 (y-1) c_p F(s,x,y) \left(6 s (y-1) y (x y (11 x y-10)+1) F(s,x,y)\right.\\ & \left.+(x y (8 x y-7)+1) F(s,x,y)^2+24 s^2 x (y-1)^2 y^3 (x y-1)\right)+(x-1) x (y (x (y (x (4 y (10 (2 x-3) (x+1) y+26 x\\ & -5)-139)+140 y+22)-20)-30 y+26)-3) F(s,x,y)^3+6 s^3 (x-1) x (y-1)^3 y^3 (y (x (y (x (y (4 (2 x-3)\\ & \left.(x+1) y+4 x+7)-12)+13 y-2)-1)-3 y+3)-1)\right)\Bigg\}, \end{aligned}
\begin{aligned}[b]\rho^{\langle\bar{s}s\bar{s}s\rangle}_{3,9;A(S)}(s) =& \int^{z_{\max}}_{z_{\min}}{\rm d}z\frac{c_1 c_p}{768 \pi } m_s^2 G(s,z) (G(s,z)+4 s (z-1) z),\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{3,9;A(S)}(s) =& \int^{z_{\max}}_{z_{\min}}{\rm d}z\Bigg\{-\frac{c_1}{2304 \pi } \left(m_s^2 \left(\left((4 z (5-6 z)+1) c_p+4 z (7-10 z)+1\right) G(s,z)-s (z-1) z \left(\left(66 z^2-58 z-2\right) c_p\right.\right.\right.\\ & \left.\left.+20 (5 z-4) z+19\right)\right)-m_c m_s \left(\left(6 c_p+13\right) G(s,z)+s \left(z \left(12 (z-1) c_p+29 z-21\right)+6\right)\right)+6 m_c^2 (2 G(s,z)\\ & +2 s (z-1) z+s)\Big)+\frac{c_2 c_p}{1024 \pi } m_s^2 (G(s,z)+2 s (z-1) z)\Bigg\}+\int^{1}_{0}dz\Bigg\{\frac{c_1}{2304 \pi } s^2 (z-1) z m_s \Big((z-1) z m_s \Big(12 (z-1) z\\\ & \left.\left.c_p+30 z^2-26 z+9\right)+(3 (z-1) z+2) m_c\right)\Bigg\}, \end{aligned}
\begin{aligned}[b] \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^2}{18432 \pi ^3 (x-1)^3 (y-1)^4} m_c \left(2 F(s,x,y) \left(m_c^2 (x y-1)^2 (y (((x-3) x+3) y+x-3)+1)\right.\right.\\ & (x (y (8 (x-3) (2 x-1) y-8 x+13)-5)-5 y+5)+3 s (x-1) (y-1) \left(y \left(22 ((x (2 (x-5) x+19)-18) x+5)\right.\right.\\ & x^2 y^5+(x (((209-56 x) x-234) x+350)-137) x y^4+(37-2 x (x (32 x+87)-36)) y^3+2 (x (x (11 x+40)\\ & \left.\left.\left.-2)-33) y^2-6 (2 x (x+2)-9) y+9 x-21\right)+3\right)\right)+3 (x-1) \left(8 ((x (2 (x-5) x+19)-18) x+5) x^2 y^5\right.\\ & +(x (((119-32 x) x-146) x+162)-55) x y^4+((x+2) ((x-26) x+4) x+15) y^3+2 (x ((5 x+2) x+14)-13)\\ & \left.y^2+2 (x-7) (x-1) y+3 (x-1)\right) F(s,x,y)^2+2 s (y-1) \left(m_c^2 (x y-1) (y (((x-3) x+3) y+x-3)+1) (y (x (y\right.\\ & (x (y (22 (x-3) (2 x-1) y-34 x+153)-12)-75 y-4)+9)+19 y-15)+3)+3 s (x-1) (y-1) y \Big(y \Big(4 ((x\\ & (2 (x-5) x+19)-18) x+5) x^2 y^5+2 \left(5-2 x^2\right) y+(x (((29-8 x) x-28) x+54)-23) x y^4+(5-4 x (3 x (x\\ & \left.\left.+3)-5)) y^3+2 (x (2 x (x+5)-5)-5) y^2+x-5\Big)+1\Big)\right)\right)\Bigg\}-\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s^2 x^2 y }{9216 \pi ^3 (x-1)^3 (y-1)^2}m_c^3\\ & (x y-1) (y (((x-3) x+3) y+x-3)+1) (y (x y (x (y (4 (x-3) (2 x-1) y-4 x+25)-4)-13 y+2)+x\\ & +3 (y-1))+1), \end{aligned}
\begin{aligned}[b] \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x}{18432 \pi ^3 (x-1)^3 (y-1)^3}m_c^2 \left((x-1) \left(c_p (x y-1) (y (((x-3) x+3) y+x-3)+1) (x\right.\right.\\ & (y (8 ((x-4) x+2) y+21 x-10)-3)-2 y+2)+8 ((x ((35 x-118) x+104)-24) x+45) x^2 y^5\\ & +40 \left(x^2 ((3 x-11) x+13)-6\right) x^3 y^6+(x (x (x (73 x+101)-669)-204)-72) x y^4+(x (((935-169 x) x\\ & \left.-310) x+252)-24) y^3+(((179-220 x) x-193) x+48) y^2+(-61 (x-1) x-36) y+x+12\right) F(s,x,y)\\ & +4 m_c^2 (x y-1) (y (((x-3) x+3) y+x-3)+1) (y (x (((5 x-6) x+2) y-4 x+3)-1)+1)+s (x-1) (y-1)\\ & \left(y \left(2 c_p (x y-1) (y (((x-3) x+3) y+x-3)+1) (x (y (11 ((x-4) x+2) y+30 x-16)-3)-2 y+2)+2 (((11\right.\right.\\ & (35 x-113) x+839) x+6) x+570) x^2 y^5+140 \left(x^2 ((3 x-11) x+13)-6\right) x^3 y^6+(x (x (x (155 x+691)\\ & -2229)-936)-204) x y^4+(x (((2353-425 x) x-92) x+492)-24) y^3+(((209-656 x) x-409) x+48) y^2\\ & \left.\left.\left.+((173-65 x) x-48) y-25 x+24\right)-6\right)\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{{\rm i} c_1 s x}{18432 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 (y (y (x ((x-3) \end{aligned}
\begin{aligned}[b]& x+3) y-3)+3)-1) \left(2 m_c^2 \left(y \left(-4 x ((5 x-6) x+2) y^2+2 ((2 x-1) x+1) y+x-2\right)+1\right)-s (x-1) x (y-1) y^2\right.\\ & \left.\left(y \left(4 c_p (((x-4) x+2) y+3 x-2)+x (2 y (25 (x (3 x-2)-2) y+130 x-21)+11)+30 y-26\right)+9\right)\right)\Bigg\}, \end{aligned}
\begin{aligned}[b] \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{3,9;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x}{36864 \pi ^3 (x-1)^3 (y-1)^3} m_c \left(3 (x-1) \left(2 (x-1) (y-1) c_p (x y (y (x (y (8 (2 x-3) (x-1) y\right.\right.\\& -2 x+21)-8)-25 y+18)-5)+3 (y-1) y+1)+8 \left(\left(x \left(6 x^2-32 x+63\right)-62\right) x+15\right) x^2 y^5+(x (((374\\ & -105 x) x-379) x+490)-137) x y^4+(((2 x (3 x-68)-165) x+20) x+23) y^3+(x ((45 x+34) x+57)-38)\\ & \left.y^2+((5 x-36) x+19) y+7 x-4\right) F(s,x,y)+m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \left(2 (x-1) (y-1) c_p (x y\right.\\ & \left.(8 x y-7)+1)+x y (x (y (8 ((6 x-25) x+9) y-65 x+223)+11)-75 y-26)+12 x+9 y-9\right)+3 s (x-1)\\ & (y-1) \left(y \left(4 (x-1) (y-1) c_p (x y (y (x (y (11 (2 x-3) (x-1) y-2 x+27)-11)-34 y+27)-8)+3 (y-1) y\right.\right.\\ & +1)+22 \left(\left(x \left(6 x^2-32 x+63\right)-62\right) x+15\right) x^2 y^5+(x ((3 (218-61 x) x-547) x+1118)-361) x y^4\\ & +(((x (2 x-341)-666) x+212) x+67) y^3+(x (19 x (6 x+17)-33)-118) y^2+(100-x (53 x+59)) y+26 x\\ & \left.\left.-41\Big)+6\right)\right)+\frac{c_2 x^2 y}{24576 \pi ^3 (x-1)^2 (y-1)^2} m_c^3 (x y-1) (y (x (y (8 ((x-1) x+1) y+26 x-17)-3)+5 y-1)-2)\Bigg\}\\ & +\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x}{36864 \pi ^3 (x-1)^3 (y-1)^2} m_c \left(m_c^2 (-(y (y (x ((x-3) x+3) y-3)+3)-1)) \left(y \left(4 (x-1) (y-1)\right.\right.\right.\\ & \left.c_p (x y (11 x y-10)+1)+x (x y (y (22 ((6 x-25) x+9) y-73 x+517)-51)-7 (29 y+2) y+30)+41 y-33\right)\\ & +6\Big)-3 s (x-1) (y-1) y \left(y \left(4 x^2 y^5 \left(2 (2 x-3) (x-1)^2 c_p+\left(x \left(6 x^2-32 x+63\right)-62\right) x+15\right)-x y^4 \left(8 (2 x-3)\right.\right.\right.\\& \left. (x-1) ((x-1) x-1) c_p+(x ((27 x-94) x+61)-182) x+63\right)+y^3 \left(9-x \left(24 (x-2) (x-1) c_p+(62 x\right.\right.\\ & \left.\left.+141) x-60\right)\right)+y^2 \left(x \left(8 (x-4) (x-1) c_p+(21 x+80) x-33\right)-18\right)+y \left(x \left(8 (x-1) c_p-17 x+2\right)+19\right)\\ & \left.\left.+3 x-10\Big)+2\right)\right)+\frac{c_2 s x^2 y^3}{24576 \pi ^3 (x-1)^2 (y-1)} m_c^3 (x (y (x (y (22 ((x-1) x+1) y+32 x-17)-47)-13 y+20)\\ & +3)+y-1)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{4;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1}{110592 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \left(-8 x^2 y^5 \left((x (x ((3 x-29) x+101)-66)-54) c_p\right.\right.\\& \left.+(((126 x-407) x+245) x+48) x+105\right)+40 ((x-3) x+3) x^3 y^6 \left(2 \left(x^2-2\right) c_p+5 x (2-3 x)+6\right)\\ & +x y^4 \left(((x ((9 x+115) x+395)-1368) x+198) c_p+((2853-x (309 x+845)) x+828) x-24\right)+y^3 \left((x (x ((31\right.\\ & \left.x-443) x+1384)-372)-6) c_p+(x ((763 x-3247) x+414)-204) x\right)+y^2 \left((((60 x-601) x+253) x+12)\right.\\ & \left.\left. c_p+((702 x-295) x+157) x\right)+y \left((x (135 x-73)-8) c_p+61 (3 x-1) x\right)-3 x c_p+2 c_p-19 x\right)\\ & +\frac{c_2 x y}{49152 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \left(c_p \left(8 ((x ((x-4) x+6)-6) x+2) x y^4+(x (x ((5 x-14) x+60)-24)-2) y^3\right.\right.\\& \left.-(x (x (x+45)-15)-4) y^2+(2 x-1) (8 x+3) y-2 x+1\right)+x y \left(40 \left(x^2 ((x-4) x+6)-2\right) x y^4+4 ((x ((34 x\right.\\& \left.-111) x+81)-16) x+10) y^3+(3 x ((36 x-109) x+61)-68) y^2+((99 x-86) x+43) y+39 x-13\right)-10 x\\ & +6 y-6\Big)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{c_1}{110592 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(6 x m_c^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \Big(y \right.\\& \left(2 c_p (((x-4) x+2) y+3 x-2)-4 (1-2 x)^2 y+9 x-4\right)-1\Big)+s (x-1) (y-1) y \left(-2 x^2 y^5 \left((x (x (7 (9 x-67) x\right.\right.\\& \left.+1471)-906)-774) c_p+(((1401 x-4367) x+1595) x+1458) x+1410\right)+140 ((x-3) x+3) x^3 y^6 \left(2 \Big(x^2\right.\\ & \left.-2\Big) c_p+5 x (2-3 x)+6\right)+x y^4 \Big((2 (x (328 x+467)-2070) x+432) c_p+((9669-x (659 x+4243)) x+3216)\\ \end{aligned}
\begin{aligned}[b] &\left.\left.x+12\Big)-y^3 \left(2 (((x (29 x+560)-1891) x+408) x+6) c_p+((5 x (1741-427 x)+708) x+252) x\right)+y^2 \left(2\right.\right.\right.\\ & \left.((11 (9 x-68) x+277) x+12) c_p+((2244 x-145) x+289) x\right)+y \left(2 (x (144 x-79)-8) c_p+(137 x-173) x\right)\\ & \left.\left.+(4-6 x) c_p+37 x\right)\right)+\frac{c_2 x y }{49152 \pi ^3 (x-1)^3 (y-1)^2}m_c^2 \left(2 m_c^2 \left(2 ((x ((x-4) x+10)-8) x+2) x y^4+(x (5 x (1-3 x).\right.\right.\\& \left.+4)-2) y^3+((x (7 x+11)-10) x+4) y^2+((3-8 x) x-3) y+x+1\right)+s (x-1) (y-1) y \left(2 c_p \left(11 ((x ((x-4) x\right.\right.\\& +6)-6) x+2) x y^4+(x (x ((8 x-23) x+87)-36)-2) y^3-(x (x (x+63)-24)-4) y^2+(2 x-1) (11 x+3) y\\ & \left.-2 x+1\right)+x \left(y \left(140 \left(x^2 ((x-4) x+6)-2\right) x y^4+2 ((((207 x-668) x+358) x+2) x+50) y^3+(x ((340 x\right.\right.\\ & \left.\left.\left.\left.-829) x+465)-180) y^2+((305 x-262) x+109) y+41 x-19\right)-8\right)-6 y+6\right)\right)\Bigg\}, \end{aligned}
(25) where the coefficient
c_p=1 for currentJ_{3,\mu\nu}^{A(S)} andc_p=-1 for currentJ_{9,\mu\nu}^{A(S)} . The spectral functions for the(0,2\{1,1\}) structure with\mathbb{C}=+1 are given as\begin{aligned}[b] \rho^{pert}_{4;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^3}{12902400 \pi ^5 (y-1)^5} \left(c_p+1\right) F(s,x,y)^3 \left(-15 (y-1) F(s,x,y) \left(7 s x y m_c m_s (11 x y (2 x y+3)\right.\right.\\ & \left.-10) (x y-1)+14 m_c^2 m_s^2 (x y (2 x y-5)+5)-6 s^2 (x-1) x^2 (y-1) y^3 (5 x y (5 x y+7)-7)\right)+21 x F(s,x,y)^2 \Big(20 s\\ & (x-1) x (y-1) y^2 (x y+2) (3 x y-1)-m_c m_s (x y-1) (x y+3) (8 x y-5)\Big)+2 (x-1) x^2 y (10 x y (3 x y+7)-49)\\ & F(s,x,y)^3-60 s x (y-1)^2 y^2 \Big(7 s x y m_c m_s (x y-1) (4 x y+5)+14 m_c^2 m_s^2 (4 x y-5)-2 s^2 (x-1) x^2 (y-1) y^3 (6 x y\\ & +7)\Big)\Big),\\ \rho^{\langle\bar{s}s\rangle}_{4;A(S)}(s) =& 0,\\ \rho^{\langle m_s\bar{s}s\rangle}_{4;A(S)}(s) =& 0,\\ \rho^{\langle g_sG^2\rangle}_{4;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^3}{15482880 \pi ^5 (x-1)^3 (y-1)^5} m_c \left(c_p+1\right) \left(6 (y-1) F(s,x,y) \left(6 s (x-1) x (y-1) y^2 m_c \Big(s x y\right.\right.\\ & (x y-1) (5 x y (5 x y+7)-7) (y (((x-3) x+3) y+x-3)+1)-7 m_s^2 (4 x y-5) (y (((x-2) x+2) y-2)+1)\Big)\\ &-7 s x y m_c^2 m_s (x y-1)^2 (11 x y (2 x y+3)-10) (y (((x-3) x+3) y+x-3)+1)-14 m_c^3 m_s^2 (x y-1) (x y (2 x y-5)\\ & +5) (y (((x-3) x+3) y+x-3)+1)-21 s^2 (x-1) x^2 (y-1) y^3 m_s (x y-1) (4 x y+5) (y (((x-3) x+3) y+x-3)\\ & +1)\Big)-21 F(s,x,y)^2 \left(2 (x-1) (y-1) m_c \left(3 m_s^2 (x y (2 x y-5)+5) (y (((x-2) x+2) y-2)+1)-10 s x^2 y^2 (x y-1)\right.\right.\\ & \left.(x y+2) (3 x y-1) (y (((x-3) x+3) y+x-3)+1)\right)+x m_c^2 m_s (8 x y-5) (x y-1)^2 (y (((x-3) x+3) y+x-3)\\ & \left.+1) (x y+3)+3 s (x-1) x (y-1) y m_s (11 x y (2 x y+3)-10) (x y-1) (y (((x-3) x+3) y+x-3)+1)\right)+(x-1)\\ & x (y (y (x ((x-3) x+3) y-3)+3)-1) F(s,x,y)^3 \left(4 x y m_c (10 x y (3 x y+7)-49)-21 m_s (x y+3) (8 x y-5)\right)\\& -6 s x (y-1)^2 y^2 m_c (y (y (x ((x-3) x+3) y-3)+3)-1) \left(7 s x y m_c m_s (x y-1) (4 x y+5)+14 m_c^2 m_s^2 (4 x y-5)\right.\\ & \left.\left.-2 s^2 (x-1) x^2 (y-1) y^3 (6 x y+7)\right)\right)+\frac{c_3}{11796480 \pi ^5 (y-1)^3} (x-1) x^2 \left(c_p+1\right) F(s,x,y)^2 \left(4 s (y-1) y (293 x y\right.\\ & \left.-27) F(s,x,y)+(181 x y-36) F(s,x,y)^2+588 s^2 x (y-1)^2 y^3\right)\Bigg\},\\ \rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{4;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^2}{512 \pi ^3 (y-1)^3} m_c \left(c_p+1\right) (x y-1) F(s,x,y) \left(s (y-1) y (5 x y-2) F(s,x,y)+(x y-1)\right. \\ & \left. F(s,x,y)^2+2 s^2 x (y-1)^2 y^3\right)\Bigg\},\\ \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{4;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x}{2304 \pi ^3 (y-1)^2} \left(c_p+1\right) \left(18 s x (y-1) y^2 F(s,x,y) \left(s (x-1) (y-1) y (5 x y-1)-m_c^2\right)\right. \end{aligned}
\begin{aligned}[b]& \left.+3 F(s,x,y)^2 \left(m_c^2 (3-3 x y)+10 s (x-1) x (y-1) y^2 (5 x y-2)\right)+2 (x-1) x y (10 x y-7) F(s,x,y)^3+6 s^3 (x-1)\right.\\ & \left. x^2 (y-1)^3 y^5\right)\Bigg\},\\ \rho^{\langle\bar{s}s\bar{s}s\rangle}_{4;A(S)}(s) = &0,\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{4;A(S)}(s) = &0,\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{4;A(S)}(s) =& 0,\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{4;A(S)}(s) = &0,\\ \rho^{\langle g_sG^2\rangle^2\;A(S)}_4(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^5 y^4 m_c^4 \left(c_p+1\right) (10 x y (3 x y+7)-49)}{46448640 \pi ^5 (x-1)^2 (y-1)^2}\\ & +\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s x^5 y^5 m_c^4 \left(c_p+1\right) (x y+2) (3 x y-1)}{1327104 \pi ^5 (x-1)^2 (y-1)},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{4;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^2}{6144 \pi ^3 (x-1)^3 (y-1)^3} m_c \left(c_p+1\right) (y (y (x ((x-3) x+3) y-3)+3)-1) \Big(3 (x-1)\\ & (x y-1) F(s,x,y)+m_c^2 (x y-1)^2+s (x-1) (y-1) y (5 x y-2)\Big)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{c_1 s x^2 y}{18432 \pi ^3 (x-1)^3 (y-1)^2}\\ & m_c \left(c_p+1\right) (y (y (x ((x-3) x+3) y-3)+3)-1) \left(m_c^2 (5 x y-2) (x y-1)+3 s (x-1) x (y-1) y^2\right)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{10;A(S)}(s) = &\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x }{27648 \pi ^3 (x-1)^2 (y-1)^2}m_c^2 \left(c_p+1\right) (x y-1)^2 \left(x y \left(20 ((x-3) x+3) y^2+26 (x-3) y\right.\right.\\ & \left.+23\Big)+18 (y-1) y+9\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{-\frac{c_1 x}{27648 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(c_p+1\right) \left(s (x-1) x (y-1) y^2 (x y (x\right.\\ & y(10 y (5 ((x-3) x+3) y-2 x+6)-9)+42 (4-5 y) y-50)+42 (y-1) y+11)-3 m_c^2 (x y-1)^2 (y (((x-3) x\\ & \left.+3) y+x-3)+1)\right)\Bigg\},\\ \end{aligned}
(26) where the coefficient
c_p=1 . The spectral functions for the(0,2\{1,1\}) structure with\mathbb{C}=-1 are given as\begin{aligned}[b]\\ \rho^{pert}_{10;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^3}{614400 \pi ^5 (y-1)^5} \left(c_p-1\right) F(s,x,y)^3 \left(5 (y-1) F(s,x,y) \left(2 s^2 (x-1) x^2 (y-1) y^3 (11 x y\right.\right.\\ & \left.-3)-5 m_c m_s (x y-1) \left(s x y (5 x y-2)-2 m_c m_s\right)\right)+x F(s,x,y)^2 \left(8 s (x-1) x (y-1) y^2 (7 x y-4)-15 m_c m_s\right.\\ & \left.(x y-1)^2\right)+4 (x-1) x^2 y (x y-1) F(s,x,y)^3+20 s x (y-1)^2 y^2 \left(5 m_c m_s \left(2 m_c m_s+s x y (1-x y)\right)+2 s^2 (x-1)\right.\\ & \left.\left.x^2 (y-1) y^3\right)\right),\\ \rho^{\langle\bar{s}s\rangle}_{10;A(S)}(s) = &0,\\ \rho^{\langle m_s\bar{s}s\rangle}_{10;A(S)}(s) = &0,\\ \rho^{\langle g_sG^2\rangle}_{10;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^3}{737280 \pi ^5 (x-1)^3 (y-1)^5} m_c \left(c_p-1\right) \left(2 (y-1) F(s,x,y) \left(2 s (x-1) x (y-1) y^2 m_c\right.\right. \\ & \left(15 m_s^2 (y (((x-2) x+2) y-2)+1)+s x y (11 x y-3) (x y-1) (y (((x-3) x+3) y+x-3)+1)\right)-5 s x y m_c^2 m_s\\ & (x y-1)^2 (5 x y-2) (y (((x-3) x+3) y+x-3)+1)+10 m_c^3 m_s^2 (x y-1)^2 (y (((x-3) x+3) y+x-3)+1)\\ & \left.-15 s^2 (x-1) x^2 (y-1) y^3 m_s (y (y (x ((x-3) x+3) y-3)+3)-1)\right)+(x y-1) F(s,x,y)^2 \left(2 (x-1) (y-1) m_c\right.\\ & \left(15 m_s^2 (y (((x-2) x+2) y-2)+1)+4 s x^2 y^2 (7 x y-4) (y (((x-3) x+3) y+x-3)+1)\right)-15 x m_c^2 m_s (x y-1)^2\\ \end{aligned}
\begin{aligned}[b] & \left.(y (((x-3) x+3) y+x-3)+1)-15 s (x-1) x (y-1) y m_s (5 x y-2) (y (((x-3) x+3) y+x-3)+1)\right)\\ & +(x-1) x (x y-1)^2 (y (((x-3) x+3) y+x-3)+1) F(s,x,y)^3 \left(8 x y m_c-15 m_s\right)+2 s x (y-1)^2 y^2 m_c (y (y (x ((x\\ & \left.-3) x+3) y-3)+3)-1) \left(-5 s x y m_c m_s (x y-1)+10 m_c^2 m_s^2+2 s^2 (x-1) x^2 (y-1) y^3\right)\right)+\frac{c_3}{2359296 \pi ^5 (y-1)^3}\\ & (x-1) x^2 \left(c_p-1\right) F(s,x,y)^2 \left(4 s (y-1) y (15 x y-2) F(s,x,y)+(9 x y-3) F(s,x,y)^2+36 s^2 x (y-1)^2 y^3\right)\Bigg\},\\ \rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{10;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^2}{512 \pi ^3 (y-1)^3} m_c \left(c_p-1\right) (x y-1) F(s,x,y) \left(s (y-1) y (5 x y-2) F(s,x,y)+(x y-1)\right.\\ & \left. F(s,x,y)^2+2 s^2 x (y-1)^2 y^3\right)\Bigg\},\\ \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{10;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x}{768 \pi ^3 (y-1)^2} \left(c_p-1\right) \left(2 s x (y-1) y^2 F(s,x,y) \left(s (x-1) (y-1) y (11 x y-3)-3 m_c^2\right)\right.\\ & +F(s,x,y)^2 \left(m_c^2 (3-3 x y)+4 s (x-1) x (y-1) y^2 (7 x y-4)\right)+4 (x-1) x y (x y-1) F(s,x,y)^3+2 s^3 (x-1) x^2\\ & \left. (y-1)^3 y^5\right)\Bigg\},\\ \rho^{\langle\bar{s}s\bar{s}s\rangle}_{10;A(S)}(s) =& 0,\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{10;A(S)}(s) =& 0,\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{10;A(S)}(s) =& 0,\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{10;A(S)}(s) =& 0,\\ \rho^{\langle g_sG^2\rangle^2}_{10;A(S)}(s) = &-\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^5 y^4 m_c^4 \left(c_p-1\right) (x y-1)}{1105920 \pi ^5 (x-1)^2 (y-1)^2}-\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{i c_1 s x^5 y^5 m_c^4 \left(c_p-1\right) (7 x y-4)}{3317760 \pi ^5 (x-1)^2 (y-1)},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{10;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^2}{6144 \pi ^3 (x-1)^3 (y-1)^3} m_c \left(c_p-1\right) (y (y (x ((x-3) x+3) y-3)+3)-1) \left(3 (x-1) (x y\right.\\ & \left.-1) F(s,x,y)+m_c^2 (x y-1)^2+s (x-1) (y-1) y (5 x y-2)\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x^2 y}{18432 \pi ^3 (x-1)^3 (y-1)^2} m_c\\ & \left(c_p-1\right) (y (y (x ((x-3) x+3) y-3)+3)-1) \left(m_c^2 (5 x y-2) (x y-1)+3 s (x-1) x (y-1) y^2\right)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x}{9216 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 \left(c_p-1\right) (x y-1) \left(y \left(x \left(y \Big(x \left(4 ((x-3) x+3) y^2-3\right)-12 y\right.\right.\right.\\ & \left.\left.\left.+18\Big)-4\right)-6 y+6\right)-3\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 x}{27648 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(c_p-1\right) \left(s (x-1) x (y-1) y^2\right.\\ & \left(x y \left(y \left(28 ((x-3) x+3) x y^2-4 (4 (x-3) x+33) y-9 x+102\right)-28\right)+30 (y-1) y+7\right)-3 m_c^2 (x y-1)^2\\ & \left.(y (((x-3) x+3) y+x-3)+1)\right)\Bigg\},\\ \end{aligned}
(27) where the coefficient
c_p=-1 . The spectral functions for the(0,2\{2,0\}) structure are given as\begin{aligned}[b] \\[-4pt]\rho^{pert}_{5,11;A(S)}(s) = &-\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^3}{6451200 \pi ^5 (y-1)^5} F(s,x,y)^3 \left(15 (y-1) F(s,x,y) \left(7 s x y m_c m_s (x y-1) \left(x y \left(11 x y \left(c_p+6\right)\right.\right.\right.\right.\\ & \left.\left.+4 c_p+99\right)-25\right)+14 m_c^2 m_s^2 \left(x y \left(x y \left(c_p+6\right)-15\right)+10\right)-s^2 (x-1) x^2 (y-1) y^3 \left(x y \left(25 x y \left(3 c_p+10\right)+28\right.\right.\\ & \left.\left.\left.\left(c_p+13\right)\right)-63\right)\right)+21 x F(s,x,y)^2 \left(m_c m_s (x y-1) \left(x y \left(4 x y \left(c_p+6\right)+2 c_p+57\right)-35\right)-2 s (x-1) x (y-1) y^2\right.\\ & \left.\left(5 x^2 y^2 \left(3 c_p+10\right)+x y \left(11 c_p+78\right)-2 \left(c_p+12\right)\right)\right)-(x-1) x^2 y \left(10 x^2 y^2 \left(3 c_p+10\right)+28 x y \left(c_p+8\right)\right.\\ \end{aligned}
\begin{aligned}[b]& \left.-7 \left(c_p+18\right)\right) F(s,x,y)^3+60 s x (y-1)^2 y^2 \left(7 s x y m_c m_s (x y-1) \left(2 x y \left(c_p+6\right)+15\right)+14 m_c^2 m_s^2 \left(2 x y \left(c_p+6\right)\right.\right.\\ & \left.\left.\left.-15\right)-2 s^2 (x-1) x^2 (y-1) y^3 \left(x y \left(3 c_p+10\right)+14\right)\right)\right),\\ \rho^{\langle\bar{s}s\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^3}{384 \pi ^3 (y-1)^4} m_c (x y-1) F(s,x,y)^2 \Big(s (y-1) y (x y (11 x y+14)-3) F(s,x,y)+(x y (x y+2)\\ & -1) F(s,x,y)^2+6 s^2 x (y-1)^2 y^3 (x y+1)\Big),\\ \rho^{\langle m_s\bar{s}s\rangle}_{5,11;A(S)}(s) =& -\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\frac{c_1 x^2}{384 \pi ^3 (y-1)^3} F(s,x,y) \left(-3 s x (y-1) y^2 F(s,x,y) \left(4 m_c^2 (x y-1)+s (x-1) (y-1) y (x y (25\right.\right.\\ & \left.x y+26)-3)\right)-2 F(s,x,y)^2 \left(m_c^2 (x y-1)^2+s (x-1) x (y-1) y^2 (x y (35 x y+39)-8)\right)-(x-1) x y (x y (5 x y+8)\\ & \left.-3) F(s,x,y)^3-12 s^3 (x-1) x^2 (y-1)^3 y^5 (x y+1)\right),\\ \rho^{\langle g_sG^2\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x^3}{7741440 \pi ^5 (x-1)^3 (y-1)^5} m_c \left(-6 (y-1) F(s,x,y) \left(-21 s^2 (x-1) x^2 (y-1) y^3 m_s (y (y (x\right.\right.\\ & ((x-3) x+3) y-3)+3)-1) \left(2 x y \left(c_p+6\right)+15\right)+s (x-1) x (y-1) y^2 m_c \left(s x y (y (y (x ((x-3) x+3) y-3)\right.\\ & \left.+3)-1) \left(x y \left(25 x y \left(3 c_p+10\right)+28 \left(c_p+13\right)\right)-63\right)-42 m_s^2 (y (((x-2) x+2) y-2)+1) \left(2 x y \left(c_p+6\right)-15\right)\right)\\ & -7 s x y m_c^2 m_s (x y-1)^2 (y (((x-3) x+3) y+x-3)+1) \left(x y \left(11 x y \left(c_p+6\right)+4 c_p+99\right)-25\right)-14 m_c^3 m_s^2 (y (y (x\\ & \left.((x-3) x+3) y-3)+3)-1) \left(x y \left(x y \left(c_p+6\right)-15\right)+10\right)\right)+(x-1) x (y (y (x ((x-3) x+3) y-3)+3)-1)\\& F(s,x,y)^3 \left(21 m_s \left(x y \left(4 x y \left(c_p+6\right)+2 c_p+57\right)-35\right)+2 x y m_c \left(-10 x^2 y^2 \left(3 c_p+10\right)-28 x y \left(c_p+8\right)\right.\right.\\ & \left.\left.+7 \left(c_p+18\right)\right)\right)+21 F(s,x,y)^2 \left(2 (x-1) (y-1) m_c \left(3 m_s^2 (y (((x-2) x+2) y-2)+1) \Big(x y \left(x y \left(c_p+6\right)-15\right)\right.\right.\\ & \left.+10\Big)-s x^2 y^2 (y (y (x ((x-3) x+3) y-3)+3)-1) \left(5 x^2 y^2 \left(3 c_p+10\right)+x y \left(11 c_p+78\right)-2 \left(c_p+12\right)\right)\right)\\& +x m_c^2 m_s (x y-1)^2 (y (((x-3) x+3) y+x-3)+1) \left(x y \left(4 x y \left(c_p+6\right)+2 c_p+57\right)-35\right)+3 s (x-1) x (y-1) y\\ & \left. m_s (y (y (x ((x-3) x+3) y-3)+3)-1) \left(x y \left(11 x y \left(c_p+6\right)+4 c_p+99\right)-25\right)\right)+6 s x (y-1)^2 y^2 m_c (y (y (x ((x\\ & -3) x+3) y-3)+3)-1) \left(7 s x y m_c m_s (x y-1) \left(2 x y \left(c_p+6\right)+15\right)+14 m_c^2 m_s^2 \left(2 x y \left(c_p+6\right)-15\right)-2 s^2\right.\\ & \left.\left.(x-1) x^2 (y-1) y^3 \left(x y \left(3 c_p+10\right)+14\right)\right)\right)\Bigg\},\\ \rho^{\langle\bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^2}{1536 \pi ^3 (y-1)^3} m_c (x y-1) F(s,x,y) \Big(3 s (y-1) y (x y(55 x y+56)-9) F(s,x,y)+4 (x y\\ & (5 x y+8)-3) F(s,x,y)^2+12 s^2 x (y-1)^2 y^3 (5 x y+4)\Big)+\frac{c_2 x^3}{1024 \pi ^3 (x-1) (y-1)^4} m_c (y (((x-2) x+2) y-2)\\ & +1) F(s,x,y) \Big(3 s (y-1) y (x y (11 x y+14)-3) F(s,x,y)+4 (x y (x y+2)-1) F(s,x,y)^2+12 s^2 x (y-1)^2 y^3\\ & (x y+1)\Big)\Bigg\},\\ \rho^{m_s\langle\bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{-\frac{c_1 x}{1152 \pi ^3 (y-1)^2} \left(-3 s x (y-1) y^2 F(s,x,y) \left(6 m_c^2 (4 x y-3)+s (x-1) (y-1) y \left(x y \left(-4 c_p\right.\right.\right.\right.\\& +125 x y+104\Big)-9\Big)\Big)-3 F(s,x,y)^2 \left(3 m_c^2 (x y-1) (2 x y-1)+s (x-1) x (y-1) y^2 \left(c_p (2-11 x y)+x y (175 x y\right.\right.\\ & \left.\left.+156)-24\right)\right)-(x-1) x y \left(-4 x y c_p+c_p+2 x y (25 x y+32)-18\right) F(s,x,y)^3-6 s^3 (x-1) x^2 (y-1)^3 y^5 (5 x y\\ & +4)\Big)+\frac{c_2 x^2 (x y-1)}{1024 \pi ^3 (x-1) (y-1)^3} \left(3 s x (y-1) y^2 F(s,x,y) \left(4 m_c^2 (x y-1)+s (x-1) (y-1) y (x y (25 x y+26)-3)\right)\right.\\ & +3 F(s,x,y)^2 \left(m_c^2 (x y-1)^2+s (x-1) x (y-1) y^2 (x y (35 x y+39)-8)\right)+2 (x-1) x y (x y (5 x y+8)-3)\\ & \left.F(s,x,y)^3+6 s^3 (x-1) x^2 (y-1)^3 y^5 (x y+1)\right)\Bigg\},\\ \end{aligned}
\begin{aligned}[b]\rho^{\langle\bar{s}s\bar{s}s\rangle}_{5,11;A(S)}(s) =& 0,\\ \rho^{\langle\bar{s}s\rangle\langle\bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) =& 0,\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^3}{4608 \pi ^3 (x-1)^3 (y-1)^4} m_c (y (y (x ((x-3) x+3) y-3)+3)-1) \left(F(s,x,y) \left(4 m_c^2 (x y (x y\right.\right.\\ & \left.+2)-1) (x y-1)+3 s (x-1) (y-1) y (x y (11 x y+14)-3)\right)+6 (x-1) (x y (x y+2)-1) F(s,x,y)^2+s (y-1) y\\& \left.\left(m_c^2 (x y (11 x y+14)-3) (x y-1)+6 s (x-1) x (y-1) y^2 (x y+1)\right)\right)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\frac{c_1 s^2 x^4 y^3}{2304 \pi ^3 (x-1)^3 (y-1)^2}\\ & m_c^3 \left(x^2 y^2-1\right) (y (y (x ((x-3) x+3) y-3)+3)-1),\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}s\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^2}{2304 \pi ^3 (x-1)^3 (y-1)^3} m_c^2 (x y-1) \left((x-1) \Big(y \Big(x \Big(y \Big(x \Big(y \Big(x \Big(10 ((x-3) x+3) y^2+26 (x\right.\\ & -3) y+23\Big)+48 y-36\Big)+7\Big)-12 y+18\Big)-3\Big)-6 y+6\Big)-3\Big) F(s,x,y)+m_c^2 (x y-1)^2 (y (((x-3) x+3) y+x\\ & -3)+1)+s (x-1) x (y-1) y^2 \left(x y \left(y \left(35 ((x-3) x+3) x y^2+(74 (x-3) x+117) y+72 x-105\right)+31\right)-12 (y\right.\\ & \left.-1) y-2\right)\Big)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x^3 y^2}{4608 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 (x y-1) (y (((x-3) x+3) y+x-3)+1) \Big(4 m_c^2 (x y\\& -1)+s (x-1) (y-1) y (x y (25 x y+26)-3)\Big)\Bigg\},\\ \rho^{\langle g_sG^2\rangle^2}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^5 y^4}{46448640 \pi ^5 (x-1)^2 (y-1)^2}m_c^4 \left(c_p (2 x y (15 x y+14)-7)+4 x y (25 x y+56)-126\right)\Bigg\}\\ & +\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x^5 y^5}{6635520 \pi ^5 (x-1)^2 (y-1)} m_c^4 \left(c_p (x y (15 x y+11)-2)+2 x y (25 x y+39)-24\right)\Bigg\},\\ \rho^{\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) = &-\int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x^2}{18432 \pi ^3 (x-1)^3 (y-1)^3} m_c (y (y (x ((x-3) x+3) y-3)+3)-1) \Big(12 (x-1) (x y (5 x y\\ & +8)-3) F(s,x,y)+4 m_c^2 (x y (5 x y+8)-3) (x y-1)+3 s (x-1) (y-1) y (x y (55 x y+56)-9)\Big)\\ & +\frac{c_2 x^3 y^2}{3072 \pi ^3 (x-1)^2 (y-1)^2} m_c^3 (x y-1) (x y (x y+2)-1)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 s x^2 y}{18432 \pi ^3 (x-1)^3 (y-1)^2} m_c (y (y (x\\ & ((x-3) x+3) y-3)+3)-1) \Big(m_c^2 (x y (55 x y+56)-9)(x y-1)+6 s (x-1) x (y-1) y^2 (5 x y+4)\Big)\\ & +\frac{c_2 s x^3 y^3}{12288 \pi ^3 (x-1)^2 (y-1)} m_c^3 (x y-1) (x y (11 x y+14)-3)\Bigg\},\\ \rho^{m_s\langle g_sG^2\rangle\langle \bar{s}\sigma\cdot Gs\rangle}_{5,11;A(S)}(s) =& \int^{x_{\max}}_{0}{\rm d}x\int^{y_{\max}}_{y_{\min}}{\rm d}y\Bigg\{\frac{c_1 x}{13824 \pi ^3 (x-1)^2 (y-1)^2} m_c^2 (x y-1) \left(y \left(-x c_p (4 x y-1) (y (((x-3) x+3) y+x-3)+1)\right.\right.\\ & \left.\left.+y \left(x \left(x \left(50 ((x-3) x+3) x y^3+6 (19 (x-3) x+32) y^2+6 (19 x-29) y+37\right)-18 (y-2)\right)-18\right)+18\right)-9\right)\\ & +\frac{c_2 x^2 y }{12288 \pi ^3 (x-1)^2 (y-1)^2}m_c^2 (x y (x y (x y (2 y (5 ((x-2) x+2) y+8 x-26)+7)+4 (8 y-5) y+7)-12 (y-1) y\\ & +3)-3)\Bigg\}+\int^{1}_{0}{\rm d}x\int^{1}_{0}{\rm d}y\Bigg\{\frac{c_1 x}{13824 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \left(3 m_c^2 (x y-1)^2 (2 x y-1) (y (((x-3) x+3) y+x-3)\right.\\ & +1)+s (x-1) x (y-1) y^2 \Big(-c_p (x y-1) (11 x y-2) (y (((x-3) x+3) y+x-3)+1)+x y \Big(y \Big(x \Big(y \Big(x \Big(175 ((x\\ & \left.-3) x+3) y^2+156 (x-3) y+12\Big)-57 y+525\Big)-202\Big)-468 y+450\Big)-120\Big)+18 (y-1) y-3\Big)\right)\\ & +\frac{c_2 x^2 y}{12288 \pi ^3 (x-1)^3 (y-1)^2} m_c^2 \Big(m_c^2 (x y-1)^2 (y (((x-2) x+2) y-2)+1)+s (x-1) x (y-1) y^2 (x y (y (x (y (35 ((x\\ & -2) x+2) y+39 x-148)+33)+78 y-62)+27)-16 (y-1) y-2)\Big)\Bigg\},\\ \end{aligned}
(28) where the coefficient
c_p=1 for currentJ_{5,\mu\nu}^{A(S)} andc_p=-1 for currentJ_{11,\mu\nu}^{A(S)} .
D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+−
- Received Date: 2022-12-28
- Available Online: 2023-05-15
Abstract: We study the mass spectra of D-wave excited