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D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+

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Zi-Yan Yang and Wei Chen. D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+[J]. Chinese Physics C. doi: 10.1088/1674-1137/acbf2c
Zi-Yan Yang and Wei Chen. D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acbf2c shu
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Received: 2022-12-28
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D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+

  • School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

Abstract: We study the mass spectra of D-wave excited csˉcˉs tetraquark states with JPC=1++ and 1+ in both symmetric 6csˉ6ˉcˉs and antisymmetric ˉ3cs3ˉcˉs color configurations using the QCD sum rule method. We construct the D-wave diquark-antidiquark type of csˉcˉs tetraquark interpolating currents in various excitation structures with (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}). Our results support the interpretation of the recently observed X(4685) resonance as a D-wave csˉcˉs tetraquark state with JPC=1++ in the (2,0{0,0}) or (0,2{2,0}) excitation mode, although some other possible excitation structures cannot be excluded exhaustively within theoretical errors. Moreover, our results provide the mass relations 6ρρ<3λλ<3λρ<3ρρ and 6ρρ<3λλ<6λλ<3ρρ for the positive and negative C-parity D-wave csˉcˉs tetraquarks, respectively. We suggest searching for these possible D-wave csˉcˉs tetraquarks in both the hidden-charm channels J/ψϕ and ηcϕ, as well as open-charm channels such as DsˉDs and DsˉDs1 .

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    I.   INTRODUCTION
    • 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 [47], doubly-charm T+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 [1117].

      In 2017, the LHCb Collaboration observed four J/ψϕ structures, i.e., X(4140), X(4274), X(4500) and X(4700), in the B+J/ψϕK+ decay process [18, 19], among which X(4140) and X(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], while X(4500) and X(4700) were new resonances. Inspired by these structures observed in the J/ψϕ invariant mass spectrum, X(4140) and X(4274) were considered to be the csˉcˉs tetraquark ground states, whereas X(4500) and X(4700) were interpreted as the csˉcˉs tetraquark excited states, in various theoretical methods [2533].

      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 four J/ψϕ states previously reported in Refs. [18, 19]. In addition, a new X(4685) state was observed in the J/ψϕ final state with 15σ significance, and its spin-parity was determined to be JP=1+. Considering its observed channel, the quantum numbers of X(4685) should be JPC=1++ with positive charge conjugation parity. Its mass and decay width are measured as m=4684±7+1316 MeV and Γ=126±15+3741 MeV. One may wonder if X(4685) and X(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 as JP=1+ for X(4685) and JP=0+ for X(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. If X(4140) and X(4274) can be assigned as the S-wave csˉcˉs tetraquark ground states with JPC=1++, the X(4685) state may be interpreted as the D-wave csˉcˉs 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-wave csˉcˉs tetraquark with JPC=1++ was calculated to be approximately 4.8 GeV. The same λ-mode excited 1++ D-wave csˉcˉs tetraquarks were also studied in the color flux-tube model with masses of approximately 4.9 and 5.2 GeV for the ˉ3cs3ˉcˉs and 6csˉ6ˉcˉs color structures, respectively [35]. In Ref. [29], the D-wave csˉcˉs 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-wave csˉcˉs tetraquarks with JPC=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 the 1++ S-wave csˉcˉs 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 axialvector 2S radial excited csˉcˉs tetraquark state.

      According to the above analyses and theoretical investigations, the newly observed X(4685) state may be explained as a ρ-mode excited D-wave csˉcˉs tetraquark with JPC=1++. In this work, we systematically study the mass spectra of the D-wave csˉcˉs with JPC=1++ and 1+ in both color symmetric 6csˉ6ˉcˉs and antisymmetric ˉ3cs3ˉcˉs 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 with JPC=1++ and 1+ 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-wave csˉcˉs tetraquark states in Sec. IV. The last section presents a summary.

    II.   INTERPOLATING CURRENTS FOR D-WAVE csˉcˉs TETRAQUARKS
    • In this section, we construct the D-wave csˉcˉs tetraquark interpolating currents with JPC=1++ and 1+. The csˉcˉs tetraquark is composed of cs diquark and ˉcˉs antidiquark fields. By analogy with the heavy baryon system, the orbital angular momentum of the tetraquark can be decomposed into L=Lρ+Lλ=lρ1+lρ2+Lλ, where lρ1(lρ2) represents the internal orbital angular momentum for the cs(ˉcˉs) field, and Lλ 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 excited csˉcˉs tetraquarks are the excitations with Lρ+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 which lρ1(lρ2) represents the internal orbital angular momentum for the cs(ˉcˉs) field, and Lλ 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 via SU(3) symmetry:

      (33)[cs](ˉ3ˉ3)[ˉcˉs]=(6ˉ3)[cs](3ˉ6)[ˉcˉs]=(6ˉ6)(ˉ33)(63)(ˉ3ˉ6)=(1827)(18)(810)(8¯10),

      (1)

      in which the color singlet structures come from the 6csˉ6ˉcˉs and ˉ3cs3ˉcˉs 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 field OS=cTaCγ5sb with JP=0+ to compose the D-wave csˉcˉs tetraquark currents by inserting covariant derivative operators. For example, one can obtain a ρ-mode P-wave diquark field with JP=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 subscripts a,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 are

      ˉOS=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 with JPC=1++ as

      JA1,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 with JPC=1+ as

      JA7,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 as 3 and 6, respectively, hereinafter. The excitation structures (Lλ,Lρ{lρ1,lρ2}), color configurations, and JPC quantum numbers for these interpolating currents are presented in Table 1. The abbreviation 3λλ/6λλ (3ρρ/6ρρ) indicates that the corresponding current contains two λ-orbital (ρ-orbital) momentums with an antisymmetric/symmetric color structure, while 3λρ/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-wave csˉcˉs 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]ˉ6JPC
      (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-wave csˉcˉs interpolating currents given by Eqs. (6) and (7).

    III.   QCD SUM RULES FOR TETRAQUARK STATES
    • 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)=id4xeiqx0|T[Jμν(x)Jρσ(0)]|0=T+μνρσΠ1(q2)+,

      (8)

      where

      T±μνρσ=(qμqνq2ηνσ±(μν))±(ρσ),ημν=qμqνq2gμν,

      (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)=Zϵμναβα(p,r)pβ,0|Jμν(x)|1(P)C(p,r)=Z+(μ(p,r)pν+ν(p,r)pμ)+Z(μ(p,r)pνν(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)=1πs<ImΠ(s)sq2iϵds,

      (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|nn|

      ρ(s)1πImΠ(s)=nδ(sM2n)0|η|nn|η|0=f2Xδ(sm2X)+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 [4042], baryons [43], and tetraquarks [4446] 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 as

      Lk(s0,M2B)=f2Xm2kHem2H/M2B=s04m2cdses/M2Bρ(s)sk,

      (13)

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

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

      (14)

      which is the function of two parameters M2B and s0. 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 constant αs. 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)=iδabˆpmc+i4gsλnab2Gnμνσμν(ˆp+mc)+(ˆp+mc)σμν12+iδab12g2sGGmcp2+mcˆp(p2m2c)4,iSabs(x)=iδab2π2x4ˆxδab12ˉss+i32π2λnab2gsGnμν1x2(σμνˆx+ˆxσμν)+δabx2192ˉsgsσGsmsδab4π2x2+iδabmsˉss48ˆximsˉsgsσGsδabx2ˆx1152,

      (15)

      where ˆp=pμγμ and ˆx=xμγμ. In this work, we evaluate the Wilson coefficients of the correlation function up to dimension ten condensates at the leading order of αs. 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.

    IV.   MASS SUM RULE ANALYSES
    • 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, 4755]:

      mc(mc)=1.27±0.02GeV,mc/ms=11.76+0.050.10,ˉqq=(0.24±0.03)3GeV3,ˉqgsσGq=M20ˉqq,ˉqqˉqq=ˉqq2,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 independent mc/ms mass ratio from PDG [3] to obtain the strange quark mass ms.

      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 mass M2B. The threshold s0 can be determined via the minimized variation of the hadronic mass mX with respect to the Borel mass M2B. The lower bound on the Borel mass M2B can be fixed by requiring a reasonable OPE convergence, while its upper bound is determined through a sufficient pole contribution. The pole contribution is defined as

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

      (17)

      where L0 is defined in Eq. (13).

      As an example, we use the color antisymmetric current JA5,μν(x) with JPC=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 mass mc. Figure 2 shows the contributions of the perturbative term and various condensate terms to the correlation function with respect to M2B when s0 tends to infinity. It is clear that the Borel mass M2B 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 mass M2B2.82GeV2. The other QCD condensates are far smaller than the quark condensate in this region of M2B. Studying the pole contribution defined in Eq. (17) reveals that the PC is very small for such D-wave csˉcˉs 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 than 20%. As a result, the reasonable Borel window for the current JA5,μν(x) is obtained as 2.94GeV2M2B3.90GeV2.

      Figure 2.  (color online) Contributions of various OPE terms to the correlation function for the current JA5,μν(x) as a function of M2B when s0.

      As mentioned previously, the variation of the extracted hadron mass mX with respect to M2B should be minimized to obtain the optimal value of the continuum threshold s0. We show the variation of mX with s0 in the left panel of Fig. 3, from which the optimized value of the continuum threshold can be chosen as s0(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 of s0 and M2B. The hadron mass for this D-wave csˉcˉs tetraquark with JPC=1++ can be obtained as

      Figure 3.  (color online) Mass curves for the interpolating current JA5,μν(x) with JPC=1++.

      mJA5=5.16+0.120.13GeV,

      (18)

      where the errors come from the uncertainties of the threshold s0, Borel mass M2B, 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 currents JS5,μν(x), JA(S)11,μν(x), and JA(S)4,μν(x) with JPC=1++ exhibit the same mass sum rule behaviors as JA5,μν(x). We present the numerical results in Table 2.

      (Lλ,Lρ{lρ1,lρ2})CurrentJPCmA/GeVs0,A/GeV2M2B,A/GeV2PCA(%)mS/GeVs0,S/GeV2M2B,S/GeV2PCS(%)
      (2,0{0,0})JA(S)1,μν1++4.70+0.120.1127(±5%)3.273.9227.34.91+0.110.1228(±5%)3.564.2026.5
      (2,0{0,0})JA(S)7,μν1+4.78+0.120.1127(±5%)3.584.1625.44.89+0.100.1128(±5%)3.604.5028.5
      (1,1{1,0})JA(S)2,μν1++4.80+0.120.1628(±5%)3.153.9439.64.84+0.120.1629(±5%)2.634.1337.9
      (1,1{1,0})JA(S)8,μν1+4.81±0.1027(±5%)3.714.5126.34.85+0.110.1028(±5%)4.695.1628.2
      (1,1{0,1})JA(S)3,μν1++4.80+0.110.1026(±5%)2.753.3126.14.82+0.120.1127(±5%)3.374.1147.0
      (1,1{0,1})JA(S)9,μν1+4.98+0.130.2326(±5%)2.733.1424.04.92+0.110.1028(±5%)3.553.9123.4
      (0,2{1,1})JA(S)4,μν1++4.80+0.100.1126(±5%)2.513.1427.54.80+0.100.1126(±5%)2.523.1527.4
      (0,2{1,1})JA(S)10,μν1+4.83+0.100.1128(±5%)3.063.8228.64.83+0.100.1228(±5%)3.083.8228.3
      (0,2{2,0})JA(S)5,μν1++5.16+0.120.1330(±5%)2.943.9041.44.69±0.0924(±5%)2.222.8227.5
      (0,2{2,0})JA(S)11,μν1+5.19+0.120.1330(±5%)3.553.9243.44.67±0.0923(±5%)2.692.8721.6
      (1,1)mixJA(S)2,μν+JA(S)3,μν1++4.80±0.1027(±5%)3.013.7624.14.93+0.090.1029(±5%)3.224.0238.4
      (1,1)mixJA(S)8,μν+JA(S)9,μν1+4.80+0.110.1326(±5%)2.713.1330.24.94±0.1029(±5%)3.374.2138.2

      Table 2.  Hadron masses of the csˉcˉs tetraquark states with different JPC 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), and JA(S)4,μν(x), the interpolating currents exhibit very different mass sum rule behaviors. As shown in the left panel for JS1,μν(x), the extracted hadron mass increases monotonically with the continuum threshold s0. Thus, one is not able to find an optimal value of s0 to minimize the variation of the hadron mass with respect to M2B. For such a situation, we define the following hadron mass ˉmX and quantity χ2(s0) to study the stability of the mass sum rules:

      ˉmX(s0)=Ni=1mX(s0,M2B,i)N,

      (19)

      χ2(s0)=Ni=1[mX(s0,M2B,i)ˉmX(s0)1]2,

      (20)

      where M2B,i(i=1,2,,N) represents N definite values for the Borel parameter M2B in the Borel window. According to the above definition, the optimal choice for the continuum threshold s0 in the QCD sum rule analysis can be obtained by minimizing the quantity χ2(s0), which is a function of only s0. This relation is shown in the right panel of Fig. 4, in which there is a minimum point at approximately s028.0GeV2. We can thus determine the working range for the continuum threshold to be s0=(28.0±1.4)GeV2, as shown in the left panel of Fig. 4. The hadron mass is thus obtained as

      Figure 4.  (color online) Mass curves (left) and χ2 curve (right) for the current JS1,μν(x) with JPC=1++.

      mJS1=4.91+0.110.12GeV.

      (21)

      In these analyses, we find that the OPE series for the JA(S)4,μν(x) and JA(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) and JS5,μν(x) with JPC=1++ agree well with the mass of the newly observed resonance X(4685), implying that X(4685) can be interpreted as a D-wave csˉcˉs tetraquark state with JPC=1++ in the excitation mode of (2,0{0,0}) or (0,2{2,0}).

      JPCS-waveP-wave
      1++Ds0ˉDs1,DsˉDs,DsˉDs1,DsˉDs1,Ds0ˉDs,Ds0ˉDs1,
      Ds1ˉDs1,Ds1ˉDs2,DsˉDs1,Ds1ˉDs1,DsˉDs2,
      J/ψϕDs1ˉDs2,hc(1P)ϕ
      1+Ds0ˉDs1,DsˉDs,DsˉDs1,DsˉDs1,Ds0ˉDs,Ds0ˉDs1,
      Ds1ˉDs1,Ds1ˉDs2,DsˉDs1,Ds1ˉDs1,DsˉDs2,
      ηcϕχc0(1P)ϕ,χc1(1P)ϕ

      Table 3.  Possible decay channels of the D-wave csˉcˉs tetraquark states with JPC=1++ and 1+.

      Considering the same physical picture for the (1,1{1,0}) and (1,1{0,1}) excitation structures, the interpolating currents JA(S)2,μν(x) and JA(S)3,μν(x) exhibit similar mass sum rules. The currents JA2,μν(x) and JA3,μν(x) give almost degenerate hadron masses, as shown in Table 2. To study their mixing effects, we also perform analyses for the mixed currents JA(S)2,μν+JA(S)3,μν. Our calculations show that the off-diagonal correlator ΠA(S)23(q2) is nonzero, implying that the currents JA2,μν(x) and JA3,μν(x) may couple to the same hadron state. The same situation arises for the interpolating currents JA(S)8,μν(x) and JA(S)9,μν(x) , which couple to the same tetraquark state.

    V.   CONCLUSION AND DISCUSSION
    • We investigated the mass spectra for the D-wave csˉcˉs tetraquark states with JPC=1++ and 1+ 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 of αs. We established reliable mass sum rules for all these currents and obtained the mass spectra of D-wave csˉcˉs tetraquarks, as shown in Table 2. Our results support the interpretation of the recently observed X(4685) structure as a D-wave csˉcˉs tetraquark state with JPC=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 [5658]. However, the situation is different for the excited csˉcˉs 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-wave csˉcˉs tetraquarks, except for those in the (0,2{2,0}) structures with two ρ-mode excitations. In Table 2, the masses for the positive C-parity tetraquarks follow the relation 6ρρ<3λλ<3λρ<3ρρ , and those for the negative C-parity tetraquarks exhibit the relation 6ρρ<3λλ<6λλ<3ρρ, which is consistent with the conclusion for P-wave ccˉcˉc 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-wave csˉcˉs tetraquarks with JPC=1++ and 1+ lie above the mass thresholds of DsˉDs, J/ψϕ, and ηcϕ. Accordingly, we present their possible decay channels in both the S-wave and P-wave in Table 3. We suggest searching for these D-wave csˉcˉs tetraquarks in both the hidden-charm channels J/ψϕ and ηcϕ , as well as open-charm channels such as DsˉDs and DsˉDs1 .

      Figure 5.  (color online) Mass spectra for the D-wave csˉcˉs tetraquark with JPC=1++ and 1+, compared with the corresponding two-meson mass thresholds.

    APPENDIX: SPECTRAL FUNCTIONS FOR D-WAVE INTERPOLATING CURRENT
    • The spectral functions for the D-wave interpolating current JA(S)i can be written as

      ρi;A(S)(s)=ρperti;A(S)(s)+ˉssρˉssi;A(S)(s)+msˉssρmsˉssi;A(S)(s)+g2sG2ρg2sG2i;A(S)(s)+ˉsσGsρˉsσGsi;A(S)(s)+msˉsσGsρmsˉsσGsi;A(S)(s)+ˉssˉssρˉssˉssi;A(S)(s)+ˉssˉsσGsρˉssˉsσGsi;A(S)(s)+g2sG2ˉssρg2sG2ˉssi;A(S)(s)+msg2sG2ˉssρmsg2sG2ˉssi;A(S)(s)+g2sG22ρg2sG22i;A(S)(s)+g2sG2ˉsσGsρg2sG2ˉsσGsi;A(S)(s)+msg2sG2ˉsσGsρmsg2sG2ˉsσGsi;A(S)(s).

      (22)

      The spectral functions for the (2,0{0,0}) structure are given as follows:

      ρpert1,7;A(S)(s)=xmax0dxymaxymindyx1612800π5(y1)5F(s,x,y)3c1(2(x1)(10((x(5(13x42)x+273)140)x+35)y228(39x245x+20)y21x(2x+5)+(10((x(2(x35)x+189)140)x+35)y2+28(x((15x74)x+70)20)y+21((23x30)x+10))cp+210)x2yF(s,x,y)3+42x((x1)(y1)(10((x(5(13x42)x+273)140)x+35)y4((x(50x+923)1165)x+590)y3+2((32x165)x+190)y2+60(x)2)y+10sx+(4((x((59x184)x+195)90)x+15)xy3+((x(58(x3)x+195)120)x+30)y2+2(3(16x45)x+110)xy60y+5(3x8)x+30)mcms+cp((x1)(y1)(10((x(2(x35)x+189)140)x+35)y2+((450x2183)x+2065)xy590y+24(23x30)x+240)sxy2+(4(x(4(x5)x+35)15)(x1)xy3+(((4(6413x)x245)x+20)x+30)y22(37x360x+30)y+55x280x+30)mcms))F(s,x,y)2+15(y1)((x1)(y1)(50((x(5(13x42)x+273)140)x+35)y414((x(10x+361)455)x+230)y3+7((79x310)x+330)y2+420(x2)y+70)s2yx2+14mcms((11((x((59x184)x+195)90)x+15)xy4+((x(2(96x325)x+765)440)x+105)y3+(x((153x485)x+430)150)y2+5((7x12)x+12)y+5(x3))sx+(6(4(2x((2x5)x+5)5)x+5)y2

      +5(7(x4)x+30)xy60y+10(x3)x+30)mcms)+cp((x1)(y1)(50((x(2(x35)x+189)140)x+35)y2+14(x((180x851)x+805)230)y+147((23x30)x+10))s2x2y3+7mcms((22(x(4(x5)x+35)15)(x1)xy32(x(((143x680)x+595)x+20)105)y2((x(359x+240)930)x+420)y+35((11x16)x+6))sxy+2(2(4(2x((x5)x+10)15)x+15)y2+5(11(x4)x+42)xy60y+10(7x9)x+30)mcms)))F(s,x,y)+60(y1)2s((x1)(y1)(4((x(5(13x42)x+273)140)x+35)y356(2(4x5)x+5)y2+21((3x10)x+10)y+35(x2))s2x2y3+2((x1)(y1)(2((x(2(x35)x+189)140)x+35)y2+14(x((8x37)x+35)10)y+7(23x30)x+70)s2x2y3+7(2(2(4(2x((x5)x+10)15)x+15)y2+5(11(x4)x+42)xy60y+10(7x9)x+30)mcmssxy(xy1)(55x280x+2(x(4(x5)x+35)15)(x1)y2+2(x((17x80)x+90)30)y+30))mcms)cpy+7mcms((4((x((59x184)x+195)90)x+15)xy4+4((x2)((17x25)x+25)x+15)y3+(x((53x210)x+250)120)y2+5((7x20)x+18)y+20x30)sxy+2(12(4(2x((2x5)x+5)5)x+5)y3+10(x(5(x4)x+24)12)y2+30((x3)x+3)y+15(x2))mcms))),ρˉss1,7;A(S)(s)=xmax0dxymaxymindyc1xmc96π3(y1)4F(s,x,y)2(2s(y1)((x1)y2((x(26x1)14)cp22x2+50x10)+11xy4(x3(cp23)2x(cp+5)+cp+7x4+24x2+1)y3((x(x(11x+37)19)7)(x1)cp2((21x40)x+28)x+7)(x1)y(7(x1)cp+10x4)x+1)F(s,x,y)+((x1)(y1)cp(2(x2+x1)xy2+x(25x)y+x+y1)+2((x((7x23)x+24)10)x+1)xy3+(x((5x9)x+7)1)y2(x1)((5x9)x+2)y(x1)2)F(s,x,y)2+6s2(y1)2y(2y3((x(x+2)4)x2cp+cp2(2(x2)x+3)x+1)+(x1)y2(2(2x2+x2)cp+x(114x)4)+2xy4(x3(cp23)2x(cp+5)+cp+7x4+24x2+1)(x1)2y(2cp+3)x+1)),ρmsˉss1,7;A(S)(s)=xmax0dxymaxymindyc196π3(y1)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((x7)x+7)2)cp+((x5)x+5)x2)6xcp+2cp+x22x+2)s(x1)x(y1)(35y4(((x((x4)x+10)8)x+2)cp+(3x((x4)x+6)8)x+2)+y3(59(x((x7)x+7)2)cp+(233x(81x+233))x118)+2y2(12((5x6)x+2)cp+(14x33)x+38)+12(x2)y+2))3s(y1)F(s,x,y)(2m2c(2y3(x4(cp+9)4x3(cp+5)+2x2(5cp+9)8x(cp+1)+2(cp+1))+4y2((x((x7)x+7)2)cp+x(x2)22)+xy(2(5x6)cp+3(x2))+y(4cp+6)+x2)s(x1)x(y1)y(25y4(((x((x4)x+10)8)x+2)cp+(3x((x4)x+6)8)x+2)+2y3(23(x((x7)x+7)2)cp+(91x(27x+91))x46)+y2(21((5x6)x+2)cp+(29x62)x+66)+12(x2)y+2))+(x1)xy(5y2(((x((x4)x+10)8)x+2)cp+(3x((x4)x+6)8)x+2)+4y(2(x((x7)x+7)2)cp3(x(x+3)3)x4)+3((5x6)x+2)cp3x+6)F(s,x,y)3+6s3(x1)x(y1)3y3(2y3(((x((x4)x+10)8)x+2)cp+(3x((x4)x+6)8)x+2)+4y2((x((x7)x+7)2)cp(x(x+4)4)x2)+xy(2(5x6)cp+3(x2))+y(4cp+6)+x2)),ρgsG21,7;A(S)(s)=xmax0dxymaxymindyx2c1mc1935360π5(x1)3(y1)5(2(x1)(2(y((x((x3)x+3)y3)y+3)1)(10((2cp+65)x470(cp+3)x3+21(9cp+13)x2140(cp+1)x+35(cp+1))y2+28(39x2+45x+(x((15x74)x+70)20)cp20)y+21((2x+5)(x)+((23x30)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), coefficient c_p=1 for current J_{1,\mu\nu}^{A(S)} while c_p=-1 for current J_{7,\mu\nu}^{A(S)} , and c_1=12,c_2=-8,c_3=4 for color antisymmetric current J_{i,\mu\nu}^{A} while c_1=24,c_2=8,c_3=20 for color symmetric current J_{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 current J_{2,\mu\nu}^{A(S)} and c_p=-1 for current J_{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 current J_{3,\mu\nu}^{A(S)} and c_p=-1 for current J_{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 current J_{5,\mu\nu}^{A(S)} and c_p=-1 for current J_{11,\mu\nu}^{A(S)} .

Reference (58)

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