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A family of double-beauty tetraquarks: Axial-vector state {{T}^{{-}}_{{bb};\overline{{u}}\overline{{s}}}}}

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S. S. Agaev, K. Azizi, B. Barsbay and H. Sundu. A family of double-beauty tetraquarks: Axial-vector state {{T}^{{-}}_{{bb};\overline{{u}}\overline{{s}}}}}[J]. Chinese Physics C. doi: 10.1088/1674-1137/abc16d
S. S. Agaev, K. Azizi, B. Barsbay and H. Sundu. A family of double-beauty tetraquarks: Axial-vector state {{T}^{{-}}_{{bb};\overline{{u}}\overline{{s}}}}}[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abc16d shu
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Received: 2020-07-13
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A family of double-beauty tetraquarks: Axial-vector state {{T}^{{-}}_{{bb};\overline{{u}}\overline{{s}}}}}

  • 1. Institute for Physical Problems, Baku State University, Az–1148 Baku, Azerbaijan
  • 2. Department of Physics, University of Tehran, North Karegar Avenue, Tehran 14395-547, Iran
  • 3. Department of Physics, Doǧuş University, Acibadem-Kadiköy, 34722 Istanbul, Turkey
  • 4. Department of Physics, Kocaeli University, 41380 Izmit, Turkey

Abstract: The spectroscopic parameters and decay channels of the axial-vector tetraquark Tbb;¯u¯s (in what follows, TAVb:¯s) are explored using the quantum chromodynamics (QCD) sum rule method. The mass and coupling of this state are calculated using two-point sum rules by taking into account various vacuum condensates, up to 10 dimensions. Our prediction for the mass of this state m=(10215±250)MeV confirms that it is stable with respect to strong and electromagnetic decays and can dissociate to conventional mesons only via weak transformations. We investigate the dominant semileptonic TAVb:¯sZ0b:¯sl¯νl and nonleptonic TAVb:¯sZ0b:¯sM decays of TAVb:¯s. In these processes, Z0b:¯s is a scalar tetraquark [bc][¯u¯s] built of a color-triplet diquark and an antidiquark, whereas M is one of the vector mesons ρ, K(892), D(2010), and Ds. To calculate the partial widths of these decays, we use the QCD three-point sum rule approach and evaluate the weak transition form factors Gi(i=0,1,2,3), which govern these processes. The full width Γfull=(12.9±2.1)×108MeV and the mean lifetime τ=5.1+0.990.71fs of the tetraquark TAVb:¯s are computed using the aforementioned weak decays. The obtained information about the parameters of TAVb:¯s and Z0b:¯s is useful for experimental investigations of these double-heavy exotic mesons.

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    I.   INTRODUCTION
    • Recently, double-beauty tetraquarks, composed of a bb diquark and a light antidiquark ¯q¯q , became a subject of intensive theoretical studies [1-6]. The interest in these states was inspired by the experimental observation of baryons Ξ++cc and measurements of their parameters [7]. The measurements were used in phenomenological models, to estimate the masses of double-beauty states [1]. These investigations demonstrated that the axial-vector tetraquark Tbb;¯u¯d (hereafter Tbb) with the mass m=(10389±12)MeV is stable with respect to strong and electromagnetic decays and can dissociate into a conventional meson only via a weak transformation. A similar conclusion about the stable nature of some tetraquarks bb¯q¯q was reached in Ref. [2] as well, where the authors of that study used methods of heavy-quark symmetry analysis.

      Double-heavy tetraquarks QQ¯q¯q, in fairness, were studied already in classical articles [8-12], in which they were examined as candidate stable four-quark compounds. The main qualitative conclusion drawn in these works was the existence of a constraint on the masses of constituent quarks. It was found that tetraquarks QQ¯q¯q may form strong-interaction stable exotic mesons, provided the ratio mQ/mq is large. Therefore, tetraquarks bb¯q¯q are the most promising candidates for stable four-quark mesons.

      Quantitative analysis of these problems continued in the following years, using the frameworks of various models and using different methods from high-energy physics. Thus, tetraquarks TQQ were explored using the chiral, dynamical, and relativistic quark models [13-17]. Axial-vector states TQQ;¯u¯d were considered in the context of the sum rule method [18,19]. Processes in which tetraquarks Tcc may be produced, namely electron-positron annihilation, heavy-ion and proton-proton collisions, and Bc meson and Ξbc baryon decays, also attracted the interest of researchers [20-24].

      The axial-vector particle Tbb was studied in our work as well [3]. We employed the quantum chromodynamics (QCD) sum rule method and evaluated the mass of this state m=(10035±260)MeV. This means that m is below both the B¯B0 and B¯B0γ thresholds; hence, this state is a strong- and electromagnetic-interaction stable tetraquark. We also explored the semileptonic decays Tbb Z0bcl¯νl, where Z0bc is the scalar tetraquark [bc][¯u¯d] composed of color-triplet diquarks, and calculated their partial widths. The predictions for the full width and mean lifetime of Tbb obtained in Ref. [3] are useful for experimental investigations of double-beauty exotic mesons.

      Other members of the bb¯q¯q family, studied in a rather detailed form, are the scalar tetraquarks Tbb;¯u¯s and Tbb;¯u¯d (in short forms, Tb:¯s and Tb:¯d, respectively). The mass and coupling of Tb:¯s and Tb:¯d were calculated in Refs. [25,26], in which we demonstrated that they cannot decay to ordinary mesons through strong and electromagnetic processes. We also investigated dominant semileptonic and nonleptonic weak decays of these tetraquarks and estimated their full width and lifetime characteristics.

      In the present article, we extend our analysis and investigate the axial-vector partner of Tb:¯s with the same quark content bb¯u¯s. It can be treated also as "s" member of the axial-vector multiplet of the states bb¯u¯q. We denote this tetraquark as TAVb:¯s and compute its spectroscopic parameters using the two-point QCD sum rule method. Calculations are performed by taking into account various vacuum condensates, up to 10 dimensions. The obtained result for its mass m=(10215±250)MeV proves that this state is stable against strong and electromagnetic decays. In fact, TAVb:¯s in the S-wave can decompose into pairs of conventional mesons BBs and B¯B0s, provided m exceeds the corresponding thresholds 10695/10692MeV, respectively. The threshold for the electromagnetic decay to the final state B¯B0sγ is 10646MeV. It is seen that even the maximal allowed value of the mass10465MeVis below all of these limits.

      Therefore, to evaluate the full width and lifetime of TAVb:¯s, we analyzed the semileptonic and nonleptonic weak decays TAVb:¯sZ0b:¯sl¯νl and TAVb:¯sZ0b:¯sM, respectively. Here, Z0b:¯s is the scalar tetraquark [bc][¯u¯s] built of a color-triplet diquark and an antidiquark, and M is one of the vector mesons ρ, K(892),  D(2010), and Ds. The weak transitions of TAVb:¯s can be described by the form factors Gi(q2) (i=0,1,2,3), which determine the differential rates dΓ/dq2 of the semileptonic and partial widths of the nonleptonic processes. These weak form factors are extracted from the QCD three-point sum rules in Section III.

      This work is structured as follows. In Section II, we calculate the mass and coupling of the tetraquarks TAVb:¯s and Z0b:¯s. For this, we derive the sum rules for their masses and couplings, by analyzing the corresponding two-point correlation functions. Numerical computations are performed by taking into account quark, gluon, and mixed condensates, up to the 10th dimension. In Section III, we compute the weak form factors Gi(q2) from the three-point sum rules for momentum transfers q2, where this method is applicable. In that section, we also determine model functions Gi(q2) and find the partial widths of the semileptonic decays TAVb:¯sZ0b:¯sl¯νl. The weak nonleptonic processes TAVb:¯sZ0b:¯sM are investigated in Section IV. This section also contains our final results for the full width and mean lifetime of the tetraquark TAVb:¯s. In Section V we discuss our obtained results and present our conclusions. Appendix contains explicit expressions of quark propagators and the correlation function used to evaluate the parameters of the tetraquark TAVb:¯s.

    II.   SPECTROSCOPIC PARAMETERS OF THE AXIAL-VECTOR TAVb:¯s AND SCALAR Z0b:¯s TETRAQUARKS
    • In this section, we calculate the mass mAV and coupling fAV of the axial-vector tetraquark TAVb:¯s, which is necessary for clarifying its nature, and conclude whether this particle is stable against strong and electromagnetic decays. Another tetraquark considered here is the scalar exotic meson Z0b:¯s that appears in the final state of the master particle's decays: spectroscopic parameters of this state enter into the expressions for the partial widths of the TAVb:¯s tetraquark's decay channels. The scalar exotic meson Z0b:¯s is a member of the bc¯q¯q family and is of interest from this perspective as well.

      The sum rules for evaluating the mass and coupling of the axial-vector tetraquark TAVb:¯s can be obtained from the two-point correlation function

      Πμν(p)=id4xeipx0|T{Jμ(x)Jν(0)}|0,

      (1)

      where Jμ(x) is the corresponding interpolating current. It is known that there are five independent diquark fields without derivatives, which can be used for formulating the current Jμ(x). Among them, scalar and axial-vector diquarks are the most stable and favorable structures for composing the tetraquark state. We suggest that TAVb:¯s is composed of the axial-vector diquark bTCγμb and the scalar antidiquark ¯uγ5C¯sT. One has to take into account that the axial-vector diquark bTCγμb has symmetric flavor but antisymmetric color organization, and its flavor-color structure is fixed as (6f,¯3c) [19]. Then, to build a color-singlet current, the light antidiquark field should belong to the triplet representation of the SUc(3) color group and has the explicit form ¯uaγ5C¯sTb¯ubγ5C¯sTa. But in calculations, owing to the symmetry constraint, it is sufficient to keep one of the light diquark terms [19]. Therefore, for the current Jμ(x) we use the following expression

      Jμ(x)=[bTa(x)Cγμbb(x)][¯ua(x)γ5C¯sTb(x)].

      (2)

      To solve the same problems in the case of the scalar tetraquark Z0b:¯s, we start from the correlation function

      Π(p)=id4xeipx0|T{JZ(x)JZ(0)}|0.

      (3)

      Here, JZ(x) is the interpolating current for Z0b:¯s

      JZ(x)=[bTa(x)Cγ5cb(x)][¯ua(x)γ5C¯sTb(x)¯ub(x)γ5C¯sTa(x)].

      (4)

      In the expressions above, a and b are the color indices, and C is the charge conjugation operator. The current (4) is composed of diquarks that belong to the triplet representation [¯3c]bc[3c]¯u¯s of the color group.

      Now, we concentrate on calculating the parameters mAV and fAV. Following the standard prescriptions of the sum rule method, we express Πμν(p) using the spectroscopic parameters of TAVb:¯s. These manipulations generate the physical or phenomenological side of the sum rules ΠPhysμν(p)

      ΠPhysμν(p)=0|Jμ|TAVb:¯s(p)TAVb:¯s(p)|Jν|0m2AVp2+.

      (5)

      Here, we isolate the ground-state contribution to ΠPhysμν(p) from the effects due to higher resonances and continuum states, which are denoted by dots. In our study, we assume that the phenomenological side of the sum rules ΠPhysμν(p) can be approximated by a zero-width single-pole term. In the case of the four-quark system, the physical side, however, also contains contributions from two-meson reducible terms [27,28]. Interaction of Jμ(x) with such a two-meson continuum generates a finite width Γ(p2) of the tetraquark and results in the following modification [29]:

      1m2p21m2p2ip2Γ(p2).

      (6)

      The contribution of the two-meson continuum can be properly taken into account by rescaling the coupling f, whereas the mass of the tetraquark m preserves its initial value [30]. These effects may be essential for strong-interaction unstable tetraquarks, because their full widths are a few100MeV. Stated differently, the two-meson continuum is important, provided the mass of the tetraquark is higher than a relevant threshold. However, even in the case of unstable tetraquarks, these effects are numerically small; therefore, it is convenient for the phenomenological side to use Eq. (5) and perform an a posteriori self-consistency check of obtained results by estimating two-meson contributions [30]. As we shall see later, the tetraquark TAVb:¯s is a strong-interaction stable particle, and mAV resides below the two-meson continuum, which justifies the zero-width single-pole approximation for ΠPhysμν(p).

      The correlator ΠPhysμν(p) can be simplified further by defining the matrix element 0|Jμ|TAVb:¯s(p)

      0|Jμ|TAVb:¯s(p)=mAVfAVϵμ,

      (7)

      where ϵμ is the polarization vector of the state TAVb:¯s. In terms of mAV and fAV , the function ΠPhysμν(p) takes the form

      ΠPhysμν(p)=m2AVf2AVm2AVp2(gμν+pμpνm2AV)+.

      (8)

      The QCD side of the sum rules can be found by substituting Jμ(x) into the correlation function (1) and contracting the relevant quark fields, which yields

      ΠOPEμν(p)=id4xeipxTr[γ5˜Sbbs(x)γ5Saau(x)]×{Tr[γν˜Sbab(x)γμSabb(x)]Tr[γν˜Saab(x)γμSbbb(x)]},

      (9)

      where Sabq(x) is the quark propagator. The propagators of heavy and light quarks used in the present work are presented in Appendix. In Eq. (9), we introduce the notation

      ˜Sq(x)=CSTq(x)C.

      (10)

      It is seen that the correlator ΠPhysμν(p) contains the Lorentz structure of the vector particle. To derive the sum rules, we choose to work with invariant amplitudes ΠPhys(p2) and ΠOPE(p2) corresponding to terms gμν, because they are free of the scalar particles' contributions.

      The sum rules for mAV and fAV can be derived by equating these two invariant amplitudes and carrying out all standard manipulations of the method. In the first stage, we apply the Borel transformation to the both sides of this equality, which suppresses the contributions of higher resonances and continuum states. In the next step, using the quark-hadron duality hypothesis, we subtract the higher resonance and continuum terms from the physical side of the equality. As a result, the sum rule equality becomes dependent on the Borel M2 and continuum threshold s0 parameters. The second equality necessary for deriving the required sum rules is obtained by applying the operator d/d(1/M2) to the first expression. Then, the sum rules for mAV and fAV are

      m2AV=Π(M2,s0)Π(M2,s0),

      (11)

      and

      f2AV=em2AV/M2m2AVΠ(M2,s0).

      (12)

      Here, Π(M2,s0) is the Borel-transformed and continuum-subtracted invariant amplitude ΠOPE(p2), and Π(M2,s0)= d/d(1/M2)Π(M2,s0). The function Π(M2,s0) has the following form:

      Π(M2,s0)=s0M2dsρOPE(s)es/M2+Π(M2),

      (13)

      where M=2mb+ms. The quantity ρOPE(s) is the two-point spectral density, whereas the second component of the invariant amplitude Π(M2) includes nonperturbative contributions calculated directly from ΠOPE(p). In the present work, we compute Π(M2,s0) by taking into account nonperturbative terms up to the 10th dimension. The explicit expression of the function Π(M2,s0) is given in Appendix.

      The sum rules for the mass mZ and coupling fZ of the scalar tetraquark Z0b:¯s can be found in the same manner. The correlator ΠPhys(p) contains only a trivial Lorentz structure proportional to I, and the relevant invariant amplitude has the simple form ΠPhys(p2)=m2Zf2Z/(m2Zp2). The QCD side of the sum rules is determined by the formula

      ΠOPE(p)=id4xeipxTr[γ5˜Saab(x)γ5Sbbc(x)]×{Tr[γ5˜Sbbs(x)γ5Saau(x)]Tr[γ5˜Sabs(x)×γ5Sbau(x)]Tr[γ5˜Sbas(x)γ5Sabu(x)]+Tr[γ5˜Saad(x)γ5Sbbu(x)]}.

      (14)

      The parameters of Z0b:¯s after evident replacements Π(M2,s0)˜Π(M2,s0) and M˜M=mb+mc+ms are determined by Eqs. (11) and (12). Here, ˜Π(M2,s0) is the transformed and subtracted invariant amplitude corresponding to the correlation function ΠOPE(p).

      The sum rules through the propagators depend on different vacuum condensates. These condensates are universal parameters of computations and do not depend on the analyzed problem. It is worth noting that the light quark propagator contains various quark, gluon, and mixed condensates of different dimensions. Some of these terms, for example, ¯qgsσGq and ¯sgsσGs, ¯qq2 and ¯ss2, ¯qqgsG2and ¯ssgsG2, and others were obtained from higher-dimensional condensates using the factorization hypothesis. However, the factorization assumption is not precise and is violated in the case of higher-dimensional condensates [31]: for the condensates of dimension 10, even the order of magnitude of such a violation is unclear. Nevertheless, the contributions of these terms are small; therefore, in what follows, we ignore the uncertainties generated by this violation. Below, we list the vacuum condensates and masses of b, c, and s quarks used in our numerical analysis:

      ¯qq=(0.24±0.01)3GeV3,¯ss=0.8 ˉqq,¯qgsσGq=m20¯qq,¯sgsσGs=m20ˉss,m20=(0.8±0.1)GeV2,αsG2π=(0.012±0.004)GeV4,g3sG3=(0.57±0.29)GeV6,ms=93+115MeV,mc=1.27±0.2GeV,mb=4.18+0.030.02GeV.

      (15)

      In Eq. (15), we introduced the following short-hand notations:

      G2=GAαβGAαβ, G3=fABCGAαβGBβδGCδα,

      (16)

      where GAαβ is the gluon field strength tensor, fABC are the structure constants of the color group SUc(3), and A,B,C=1,2,...8.

      The mass and coupling of the tetraquarks (11) and (12) also depend on the Borel and continuum threshold parameters M2 and s0. The M2 and s0 are the auxiliary quantities, and their correct choice is one of the important problems in sum rule studies. Proper working regions for M2 and s0 must satisfy restrictions imposed on the pole contribution (PC) and convergence of the operator product expansion measured by the ratio R(M2), which we define respectively by the expressions

      PC=Π(M2,s0)Π(M2,),

      (17)

      and

      R(M2)=ΠDimN(M2,s0)Π(M2,s0).

      (18)

      Here, ΠDimN(M2,s0) is a contribution to the correlation function of the last term (or sum of the last few terms) in the operator product expansion. In the present work, we use the following restrictions imposed on these parameters: at the maximal edge of M2, the pole contribution should obey PC>0.2, and at the minimum of M2, we require fulfilment of R(M2)0.01. Lets us note that we estimate R(M2) using the last three terms in the OPE DimN=Dim(8+9+10).

      Variations of M2 and s0 within the allowed working regions are the main sources of theoretical errors in sum rule computations. Therefore, the Borel parameter M2 should be fixed for minimizing the dependence of extracted physical quantities on its variations. The situation with s0 is more subtle, because it bears physical information about the excited states of the tetraquark TAVb:¯s. In fact, the continuum threshold parameter s0 separates the ground-state contribution from the ones of higher resonances and continuum states; hence, s0 should be below the first excitation of TAVb:¯s. However, available information on the excited states of tetraquarks is limited to only a few theoretical studies [32-34]. As a result, one fixes s0 to achieve maximalPC, ensuring fulfilment of the other constraints and simultaneously keeping the computation self-consistency under control. The latter means that the gap s0mAV in the case of heavy tetraquarks should be 600MeV, which serves as a measure of excitation.

      Numerical analysis suggests that regions

      M2[9,12] GeV2, s0[115,120] GeV2,

      (19)

      satisfy all of the aforementioned constraints on M2 and s0. Thus, at M2=12GeV2, the pole contribution is 0.23, and at M2=9GeV2, it amounts to 0.62. These values of M2 limit the boundaries of a region in which the Borel parameter can be changed. At the minimum of M2=9GeV2, we get R0.005. In addition, at the minimum of the Borel parameter, the perturbative contribution is 79% of the result overshooting the nonperturbative effects.

      For mAV and fAV, we have obtained

      mAV=(10215±250)MeV,fAV=(2.26±0.57)×102GeV4.

      (20)

      In Eq. (20), the theoretical uncertainties of computations are shown as well. For the mass mAV, these uncertainties are ±2.4% of the central value, and for the coupling fAV, they amount to ±25%, but in both cases, they remain within the limits accepted by the sum rule computations. In Fig. 1 , we plot our prediction for mAV as a function of M2 and s0: one can see a mild dependence of mAV on these parameters. It is also evident that

      Figure 1.  (color online) Dependence of the mass mAV on the Borel M2 (left panel) and continuum threshold s0 parameters (right panel).

      s0mAV=[510,740]MeV,

      (21)

      which is a reasonable mass gap between the ground-state and excited heavy tetraquarks.

      Returning to the issue of the two-meson continuum, we can now compare the mass of the tetraquark TAVb:¯s with the energy level of this continuum. It is clear that the two-meson continuum may be populated by pairs BBs and B¯B0s, and that TAVb:¯s is 480MeV below it. This difference is comparable to (21); hence, one can ignore the two-meson continuum's impact on the physical parameters of TAVb:¯s.

      The mass mZ and coupling fZ of the state Z0b:¯s are found from the sum rules by utilizing the following working windows for M2 and s0

      M2[5.5,6.5]GeV2, s0[52,54]GeV2.

      (22)

      The regions (22) satisfy standard restrictions associated with the sum rule computations. In fact, at M2=5.5GeV2, the ratio R is 0.009; hence, the convergence of the sum rules is satisfied. The pole contribution PC at M2=6.5GeV2 and M2=5.5GeV2 equals to 0.23 and 0.61, respectively. At the minimum of M2, the perturbative contribution constitutes 72% of the entire result and considerably exceeds that of nonperturbative terms.

      For mZ and fZ, our computations yield

      mZ=(6770±150)MeV,fZ=(6,3±1.3)×103GeV4.

      (23)

      In Fig. 2, we depict the mass of the tetraquark Z0b:¯s and demonstrate its dependence on M2 and s0.

      Figure 2.  (color online) The mass mZ of the tetraquark Z0b:¯s as a function of the parameters M2 (left panel) and s0 (right panel).

      The mass of the axial-vector tetraquark TAVb:¯s was calculated in Ref. [19] in the context of the QCD sum rule method, using different interpolating currents. Computations were performed with dimension 8 accuracy, and two lowest predictions for the mass of the axial-vector particle bb¯q¯s were obtained within ranges (10300±300)MeV and (10300±400)MeV. Our result is close to the central value of these predictions. The difference in theoretical errors can be attributed to the higher accuracy of our computations and more detailed quark propagators used in analysis. The authors of Ref. [19] noted the strong interaction stable nature of TAVb:¯s. As we will see below, our investigation proves that TAVb:¯s is stable against strong and radiative decays and can transform only through weak processes.

      The scalar tetraquark with the quark content [bc][¯u¯s] was explored recently in Ref. [35]. The predicted mass of this state (7.14±0.12)GeV obtained there is larger than our prediction (23). Such a sizeable difference between the two results can be explained by some factors. Thus, in the present work, calculations have been performed by taking into account dimension  10 condensates, whereas in Ref. [35], the authors included nonperturbative terms up to the eighth dimension into analysis. We have used more detailed expressions for quark propagators, including the terms g2s¯qq2 and ¯qqg2sG2 in the light and g3sG3 in the heavy quark propagators. However, in our view, the choice of the working windows for the parameters M2 and s0 is the main source of fixed discrepancies. The regions for M2 and s0 should be extracted from the analysis of constraints (17) and (18) imposed on the invariant amplitude Π(M2,s0). The PC in the present investigation varies within 0.61-0.23, which corresponds to the boundaries of the Borel region. Let us emphasize that we extract the parameters mZ and fZ approximately in the middle region of the window (22), where the pole contribution is PC0.420.45. The working regions for M2 and s0 used in Ref. [35] ensure only PC0.31, which may generate differences in the extracted values of mZ.

    III.   WEAK FORM FACTORS Gi(p2) AND SEMILEPTONIC DECAYS TAVb:¯sZ0b:¯sl¯νl
    • The analysis performed in the previous section confirms that the tetraquark TAVb:¯s is stable against the strong and electromagnetic decays. Indeed, the mass of this state mAV=10215MeV is 480/477MeV below the thresholds 10695/10692MeV for its strong decays to mesons BBs and B¯B0s, respectively. The maximum of the mass 10465MeV is still below these limits. The threshold 10646MeV for the process TAVb:¯sB¯B0sγ also exceeds the maximal allowed value of mAV , which forbids this electromagnetic decay. Therefore, the full width and mean lifetime of TAVb:¯s are determined by its weak decays.

      There are different weak decay channels of TAVb:¯s, which can be generated by sub-processes bWc and bWu. The decays triggered by the transition bWc are dominant processes relative to the ones connected with bWu: the latter decays are suppressed relative to the dominant decays by a factor |Vbu|2/|Vbc|2 0.01, with Vq1q2 being the Cabibbo-Khobayasi-Maskawa (CKM) matrix elements. In the present work, we restrict ourselves to the analysis of the dominant weak decays of TAVb:¯s (see Fig. 3).

      Figure 3.  (color online) The Feynman diagram for the semileptonic decay TAVb:¯sZ0b:¯sl¯νl. The black square denotes the effective weak vertex.

      The dominant processes themselves can be categorized into two groups: the first group contains the semileptonic decays TAVb:¯sZ0b:¯sl¯νl, whereas the nonleptonic transitions TAVb:¯sZ0b:¯sM belong to the second group. In this section, we consider the semileptonic decays and calculate the partial widths of the processes TAVb:¯sZ0b:¯sl¯νl, where l is one of the lepton species e,μ and τ. Owing to the large mass difference between the initial and final tetraquarks, 3445MeV, all of these semileptonic decays are kinematically allowed ones.

      The effective Hamiltonian to describe the subprocess bWc at the tree-level is given by the expression

      Heff=GF2Vbc¯cγμ(1γ5)b¯lγμ(1γ5)νl,

      (24)

      with GF and Vbc being the Fermi coupling constant and CKM matrix element, respectively. A matrix element of Heff between the initial and final tetraquarks is equal to

      Z0b:¯s(p)|Heff|TAVb:¯s(p)=LμHμ,

      (25)

      where Lμ and Hμ are the leptonic and hadronic factors, respectively. A treatment of Lμ is trivial; therefore, we consider the matrix element Hμin a detailed form, which depends on the parameters of the tetraquarks. After factoring out the constant factors, Hμ is the matrix element of the current

      Jtrμ=¯cγμ(1γ5)b.

      (26)

      The matrix element Z0b:¯s(p)|Jtrμ|TAVb:¯s(p) describes the weak transition of the axial-vector tetraquark to the scalar particle and is expressible in terms of four weak form factors Gi(q2) that parametrize long-distance dynamical effects of this transformation [36,37]

      Z0b:¯s(p)|Jtrμ|TAVb:¯s(p)=˜G0(q2)ϵμ+˜G1(q2)(ϵp)Pμ+˜G2(q2)(ϵp)qμ+i˜G3(q2)εμναβϵνpαpβ.

      (27)

      The scaled functions ˜Gi(q2) are connected with the dimensionless form factors Gi(q2) by the equalities

      ˜G0(q2)=˜mG0(q2), ˜Gj(q2)=Gj(q2)˜m, j=1,2,3.

      (28)

      Here, ˜m=mAV+mZ, pμ and ϵμ are the momentum and polarization vector of the tetraquark TAVb:¯s, p is the momentum of the scalar state Z0b:¯s. We use also Pμ=pμ+pμand qμ=pμpμ, the latter being the momentum transferred to the leptons. It is evident that q2 varies within m2lq2(mAVmZ)2, where ml is the mass of a lepton l.

      The sum rules for the form factors Gi(q2) can be obtainedby analyzing the three-point correlation function

      Πμν(p,p)=i2d4xd4yei(pypx)×0|T{JZ(y)Jtrν(0)Jμ(x)}|0.

      (29)

      To this end, we have to express Πμν(p,p) using the masses and couplings of the tetraquarks and thus determine the physical side of the sum rules ΠPhysμν(p,p). The function ΠPhysμν(p,p) can be presented as

      ΠPhysμν(p,p)=0|JZ|Z0b:¯s(p)Z0b:¯s(p)|Jtrν|TAVb:¯s(p,ϵ)(p2m2AV)(p2m2Z)×TAVb:¯s(p,ϵ)|Jμ|0+,

      (30)

      where we take into account the contribution of the ground-state particles and denote the effects of the excited and continuum states by dots.

      The phenomenological side of the sum rules can be simplified by substituting into Eq. (30) the expressions of matrix elements in terms of the tetraquarks' masses and couplings as well as weak transition form factors. For these purposes, we employ Eqs. (7) and (27) and define the matrix element of Z0b:¯s

      0|JZ|Z0b:¯s(p)=fZmZ.

      (31)

      Then, one gets

      ΠPhysμν(p,p)=fAVmAVfZmZ(p2m2AV)(p2m2Z)×{˜G0(q2)(gμν+pμpνm2AV)+[˜G1(q2)Pμ+˜G2(q2)qμ](pν+m2AV+m2Zq22m2AVpν)i˜G3(q2)εμναβpαpβ}+.

      (32)

      We should also calculate the correlation function in terms of the quark propagators and find ΠOPEμν(p,p). The function ΠOPEμν(p,p) is the second side of the sum rules and has the following form

      ΠOPEμν(p,p)=d4xd4yei(pypx){Tr[γ5˜Sbas(xy)×γ5Sabu(xy)](Tr[γμ˜Saab(yx)γ5Sbic(y)γν(1γ5)×Sibb(x)]+Tr[γμ˜Siab(x)(1γ5)γν˜Sbic(y)γ5×Sabb(yx)])Tr[γ5˜Sbas(xy)γ5Sabu(xy)]×(Tr[γμ˜Saab(yx)γ5Sbic(y)γν(1γ5)Sibb(x)]+Tr[γμ˜Siab(x)(1γ5)γν˜Sbic(y)γ5Sabb(yx)])}.

      (33)

      To extract expressions of the form factors Gi(q2), we equate invariant amplitudes corresponding to the same Lorentz structures both in ΠPhysμν(p,p) and ΠOPEμν(p,p), carry out double Borel transformations over the variables p2 and p2, and perform continuum subtraction. For instance, to extract the sum rule for ˜G0(q2), we use the structure gμν, whereas for ˜G3(q2), the term εμναβpαpβ can be employed. The sum rules for the scaled form factors ˜Gi(q2) can be written in a single formula

      ˜Gi(M2,s0,q2)=1fAVmAVfZmZs0M2dse(m2AVs)/M21×s0˜M2dsρi(s,s)e(m2Zs)/M22,

      (34)

      where ρi(s,s) are spectral densities computed as the imaginary parts of the corresponding terms in ΠOPEμν(p,p). They contain perturbative and nonperturbative contributions and are found in the present work with dimension-6 accuracy. In Eq. (34), M2=(M21,M22) and s0=(s0, s0) are the Borel and continuum threshold parameters, respectively. The pair of parameters (M21, s0) corresponds to the initial tetraquark's channels, whereas (M22, s0) describes the final-state tetraquark.

      As usual, the form factors ˜Gi(M2,s0,q2) contain various input parameters, which should be determined before numerical analysis. The vacuum condensates of quark, gluon, and mixed operators are already presented in Eq. (15). The masses and couplings of the tetraquarks TAVb:¯s and Z0b:¯s have been extracted in Section II. The Borel and continuum threshold parameters M2 and s0 should be chosen so as to meet all restrictions of sum rule computations. One has also to bear in mind that ˜Gi(M2,s0,q2) depends on the masses and couplings of the initial and final tetraquarks, which have been evaluated also in the context of the sum rule approach. We fix the auxiliary parameters (M21, s0) and (M22, s0) as in the corresponding mass computations, because they satisfy standard constraints of three-point sum rule calculations and do not generate additional uncertainties in the spectroscopic parameters of relevant tetraquarks.

      The form factors ˜Gi(q2) determine the differential decay rate dΓ/dq2 of the semileptonic decay TAVb:¯sZ0b:¯sl¯νl, the explicit expression of which can be found in Ref. [3]. The partial width of the process is equal to an integral of this rate over the momentum transfer q2 within the limits m2lq2(mAVmZ)2. Our results for the form factors are plotted in Fig. 4. The QCD sum rules lead to reliable predictions at m2lq28GeV2. However, these predictions do not cover the entire integration region m2lq211.87GeV2. To solve this problem, one has to introduce extrapolation functions Gi(q2) of a relatively simple analytic form, which, at q2 accessible to the QCD sum rules, coincide with their predictions but can be used in the entire region.

      Figure 4.  (color online) Sum rule results for the form factors G0(q2) (red circles) and G1(q2) (blue squares). The solid curves are fit functions G0(q2) and G1(q2).

      For these purposes, we choose to work with the functions

      Gi(q2)=Gi0exp[gi1q2m2AV+gi2(q2m2AV)2],

      (35)

      in which the parametersGi0,gi1, and gi2 should be fitted to satisfy the sum rules' predictions. The parameters of the functions Gi(q2) , obtained numerically, are presented in Table 1. The functions Gi(q2) are also shown in Fig. 4 : one can see a good agreement between the sum rule predictions and fit functions.

      Gi(q2) Gi0 gi1 gi2
      G0(q2) 4.91 19.29 15.34
      G1(q2) 2.94 18.73 20.09
      G2(q2) 22.67 20.50 22.95
      G3(q2) 21.14 20.77 23.62

      Table 1.  Parameters of the extrapolating functions Gi(q2).

      In the numerical computations for the Fermi constant, CKM matrix elements, and masses of leptons, we use

      GF=1.16637×105GeV2,|Vbc|=(42.2±0.08)×103,

      (36)

      me=0.511MeV, mμ=105.658MeV, and mτ= (1776.82±0.16)MeV [38]. The predictions obtained for the partial widths of the semileptonic decays TAVb:¯sZ0b:¯sl¯νl are written as

      Γ(TAVb:¯sZ0b:¯se¯νe)=(5.34±1.43)×108MeV,Γ(TAVb:¯sZ0b:¯sμ¯νμ)=(5.32±1.41)×108MeV,Γ(TAVb:¯sZ0b:¯sτ¯ντ)=(2.15±0.54)×108MeV,

      (37)

      and are main results of this section.

    IV.   NONLEPTONIC DECAYS TAVb:¯sZ0b:¯sM
    • The second class of the weak decays of the tetraquark TAVb:¯s are the processes TAVb:¯sZ0b:¯sM, which may affect the full width and lifetime of the tetraquark TAVb:¯s. Here, we study the nonleptonic weak decays TAVb:¯sZ0b:¯sM of the tetraquark TAVb:¯s in the framework of the QCD factorization method. This approach was applied for investigating the nonleptonic decays of conventional mesons [39,40] but can be also used for investigating the decays of tetraquarks. Thus, the nonleptonic decays of scalar exotic mesons Tb:¯s, Tb:¯s, Z0bc, and Tbs;¯u¯d were explored using this approach in Refs. [25,26,41,42], respectively. The weak decays of double- and fully-heavy tetraquarks were analyzed in Refs. [43,44].

      We consider processes where M is one of the vector mesons ρ, K(892),  D(2010), and Ds. We provide details of analysis for the decay TAVb:¯sZ0b:¯sρ and write the final predictions for other channels. The relevant Feynman diagram is shown in Fig. 5.

      Figure 5.  (color online) The diagram for the nonleptonic decay TAVb:¯sZ0b:¯sρ.

      At the quark level, the effective Hamiltonian for this decay is given by the expression

      Heffn.lep=GF2VbcVud[c1(μ)Q1+c2(μ)Q2],

      (38)

      where

      Q1=(¯diui)VA(¯cjbj)VA,Q2=(¯diuj)VA(¯cjbi)VA,

      (39)

      i and j are the color indices, and (¯q1q2)VA means

      (¯q1q2)VA=¯q1γμ(1γ5)q2.

      (40)

      The short-distance Wilson coefficients c1(μ) and c2(μ) are given on the factorization scale μ.

      In the factorization method, the amplitude of the decay TAVb:¯sZ0b:¯sρ has the form

      A=GF2VbcVuda(μ)ρ(q)|(¯diui)VA|0×Z0b:¯s(p)|(¯cjbj)VA|TAVb:¯s(p),

      (41)

      where

      a(μ)=c1(μ)+1Ncc2(μ),

      (42)

      with Nc=3 being the number of quark colors. The only unknown matrix element ρ(q)|(¯diui)VA|0 in A can be defined in the following form

      ρ(q)|(¯diui)VA|0=fρmρϵμ(q).

      (43)

      Then, it is evident that

      A=iGF2fρVbcVuda(μ)[˜G0(q2)ϵμ(p)ϵμ(q)+2˜G1(q2)(pϵ(p))(pϵ(q))+i˜G3(q2)εμναβϵμ(q)ϵν(p)pαpβ].

      (44)

      The decay modes TAVb:¯sZ0b:¯sK(892)[D(2010), Ds] can be analyzed in a similar way. To this end, we have to replace in relevant expressions the spectroscopic parameters (mρ,fρ) of the ρ meson with masses and decay constants of the mesons K(892), D(2010), and Ds and make the substitutions VudVus, Vcd, and Vcs.

      The width of the nonleptonic decay TAVb:¯sZ0b:¯sρ can be evaluated using the expression

      Γ=|A|224πm2AVλ(mAV,mZ,mρ),

      (45)

      where

      λ(a,b,c)=12a[a4+b4+c42(a2b2+a2c2+b2c2)]1/2.

      (46)

      The key component in Eq. (45), i.e., |A|2 has a simple form

      |A|2=j=0,1,2Hj˜G2j+H3˜G0˜G1,

      (47)

      where Hj are given by the expressions

      H0=m4ρ+(m2AVm2Z)2+2m2ρ(5m2AVm2Z)4m2ρm2AV,H1=[m4ρ+(m2AVm2Z)22m2ρ(m2AV+m2Z)]24m2ρm2AV,H2=12[m4ρ+(m2AVm2Z)22m2ρ(m2AV+m2Z)],H3=12m2ρm2AV[m6ρ+(m2AVm2Z)3m4ρ(m2AV+3m2Z)m2ρ(m4AV+2m2Zm2AV3m2Z)].

      (48)

      In Eq. (47), we take into account that the weak form factors ˜Gj are real functions of q2, and their values for the process TAVb:¯sZ0b:¯sM are fixed at q2=m2M.

      All input information necessary for numerical analysis is presented in Table 2: the table lists the spectroscopic parameters of the final-state mesons and the CKM matrix elements. For the masses of the vector mesons, we use information from PDG [38]. The decay constants of mesons ρ and K(892) are also taken from this source. The decay constants of mesons D and Ds are theoretical predictions obtained in the lattice QCD framework [45]. The coefficients c1(mb), and c2(mb) with next-to-leading order QCD corrections have been borrowed from Refs. [46-48]

      Quantity Value
      mρ (775.26±0.25)MeV
      mK (891.66±0.26)MeV
      mD (2010.26±0.05)MeV
      mDs (2112.2±0.4)MeV
      fρ (210±4)MeV
      fK (204±7)MeV
      fD (223.5±8.4)MeV
      fDs (268.8±6.6)MeV
      |Vud| 0.97420±0.00021
      |Vus| 0.2243±0.0005
      |Vcd| 0.218±0.004
      |Vcs| 0.997±0.017

      Table 2.  Masses and decay constants of the final-state vector mesons and CKM matrix elements.

      c1(mb)=1.117, c2(mb)=0.257.

      (49)

      For the decay TAVb:¯sZ0b:¯sρ , our calculations yield

      Γ(TAVb:¯sZ0b:¯sρ)=(3.47±0.92)×1010MeV.

      (50)

      The partial widths of the remaining three nonleptonic decays are presented below

      Γ(TAVb:¯sZ0b:¯sK(892))=(1.47±0.37)×1011MeV,Γ(TAVb:¯sZ0b:¯sD(2010))=(1.54±0.39)×1011MeV,Γ(TAVb:¯sZ0b:¯sDs)=(4.97±1.32)×1010MeV.

      (51)

      It is evident that the parameters of the processes TAVb:¯s Z0b:¯sρ and TAVb:¯sZ0b:¯sDs are comparable to each other and may affect predictions for the tetraquark TAVb:¯s: the other two decays can be safely neglected in the computation of Γfull and τ. Then, using Eqs. (37), (50), and (51), we find

      Γfull=(12.9±2.1)×108MeV,τ=5.1+0.990.71×1015s,

      (52)

      which are principally new predictions of the present article.

    V.   DISCUSSION AND CONCLUDING NOTES
    • We have calculated the mass, width, and lifetime of the stable axial-vector tetraquark TAVb:¯s with the content bb¯u¯s. This particle is a strange partner of the tetraquark Tbb, which was explored in Ref. [3]. The width and lifetime of Tbb

      ˜Γfull=(7.17±1.23)×108MeV,˜τ=9.18+1.901.34×1015s,

      (53)

      are comparable to those of the tetraquark TAVb:¯s.

      The tetraquark TAVb:¯s is the last of the four scalar and axial-vector states bb¯u¯s and bb¯u¯d considered in our works. The spectroscopic parameters and widths of the scalar tetraquarks Tb:¯s and Tb:¯d were calculated in Refs. [25,26]. We demonstrated there that Tb:¯s and Tb:¯d are stable against the strong and electromagnetic decays, and using the dominant semileptonic and nonleptonic decay channels of these particles, we estimated their full widths and lifetimes. The information about the tetraquarks composed of a heavy diquark bb and light antidiquarks is presented in Table 3.

      Tetraquark (JP) Mass/MeV Width/MeV Lifetime
      TAVb:¯s(1+) 10215±250 (12.9±2.1)×108 5.1+0.990.71 fs
      Tbb(1+) 10035±260 (7.17±1.23)×108 9.18+1.901.34 fs
      Tb:¯s(0+) 10250±270 (15.21±2.59)×1010 0.433+0.0890.063 ps
      Tb:¯d(0+) 10135±240 (10.80±1.88)×1010 0.605+0.1260.089 ps

      Table 3.  Parameters of the scalar and axial-vector tetraquarks composed of the diquark bb and light antidiquarks.

      It is seen that the scalar particles are heavier than their axial-vector counterparts: This mass for tetraquarks bb¯u¯s is equal to 35MeV, and for particles with quark content bb¯u¯d, it reaches100MeV. It is also clear that the mass splitting of the strange and nonstrange axial-vector tetraquarks, 180MeV, exceeds the value of the same parameter for the scalars, 115MeV. These estimates are obtained using the central values of various tetraquarks' masses calculated using the QCD sum rule method. It is known that this method is prone to theoretical uncertainties; therefore, mass splitting between double-beauty tetraquarks and hierarchy of the particles outlined here must be considered with some caution. Nevertheless, we hope that the picture described above is a quite reliable image of the real situation.

      The widths and lifetimes of these tetraquarks have yielded important insights into their dynamical properties. It is worth noting that the semileptonic decay channels crucially affect the full widths of these tetraquarks: our investigations have shown that the partial width of the semileptonic decay is enhanced relative to the nonleptonic one by 23 orders of magnitude. The widths of the scalar tetraquarks Tb:¯s and Tb:¯d are considerably smaller than the widths of the axial-vector particles Tbb and TAVb:¯s. As a result, the mean lifetimes of the scalar tetraquarks are ~1ps, whereas for the axial vector states, we get τ10 fs. Stated differently, the scalar tetraquarks Tb:¯s and Tb:¯d are heavier and live longer than the corresponding axial-vector particles.

      The spectroscopic parameters and lifetimes of the axial-vector states Tbb and TAVb:¯s were also explored in Refs. [1,5]. The lifetime 367fs of the state Tbb  predicted in Ref. [1] is considerably longer than our result 9.18fs. The lifetimes τ800fs of the tetraquarks Tbb and TAVb:¯s obtained in Ref. [5] exceed our predictions as well. Let us note that, in Ref. [5], the authors considered only nonleptonic decays of the axial-vector tetraquarks. We have reevaluated the lifetime of TAVb:¯s using Eqs. (50) and (51) and found τ753fs. Despite the fact that the channels that were explored in Ref. [5] differ from the decays that were considered in the present work, for τ, they lead to compatible predictions. One of the reasons is that, in both cases, the amplitudes of the nonleptonic weak decays contain two CKM matrix elements, which suppress their partial widths and branching ratios relative to the semileptonic channels. Evidently, our results for the nonleptonic decays of TAVb:¯s can be refined by including into analysis some relevant channels from Ref. [5]. However, for discovering stable exotic mesons, their semileptonic decays seem to be more promising than other processes.

    APPENDIX A: THE PROPAGATORS Sq(Q)(x) AND INVARIANT AMPLITUDE Π(M2,s0)
    • In the present work, we use the light quark propagator Sabq(x), which is given by the following formula

      Sabq(x)=iδabx2π2x4δabmq4π2x2δab¯qq12+iδabxmq¯qq48δabx2192¯qgsσGq+iδabx2xmq1152¯qgsσGqigsGαβab32π2x2[xσαβ+σαβx]iδabx2xg2s¯qq27776δabx4¯qqg2sG227648+.

      For the heavy quarks Q , we utilize the propagator SabQ(x)

      SabQ(x)=id4k(2π)4eikx{δab(k+mQ)k2m2QgsGαβab4σαβ(k+mQ)+(k+mQ)σαβ(k2m2Q)2+g2sG212δabmQk2+mQk(k2m2Q)4+g3sG348δab(k+mQ)(k2m2Q)6[k(k23m2Q)+2mQ(2k2m2Q)](k+mQ)+}.

      Above, we have used the notation

      GαβabGαβAtAab,  

      where GαβA is the gluon field strength tensor, and tA=λA/2 with λA being the Gell-Mann matrices, A=1,2,,8.

      The invariant amplitude ΠOPE(p2) used for calculating the mass and coupling of the tetraquark Tb:¯s after the Borel transformation and subtraction procedures takes the following form

      Π(M2,s0)=s0M2dsρOPE(s)es/M2+Π(M2),

      where

      ρOPE(s)=ρpert.(s)+8N=3ρDimN(s),  Π(M2)=10N=6ΠDimN(M2).

      Components of the spectral density are given by the formulas

      ρ(s)=10dα1a0dβρ(s,α,β),  ρ(s)=10dαρ(s,α),

      depending on whether ρ(s,α,β) is a function of α and β or only α. The same is true also for terms Π(M2), i.e.,

      ΠDimN(M2)=10dα1a0dβΠDimN(M2,α,β),Π(M2)=10dαΠDimN(M2,α).

      In these expressions, α and β are Feynman parameters.

      The perturbative and nonperturbative contributions of dimensions 3, 4, and 5 are terms of (A6) types. For relevant spectral densities, we get

      ρpert.(s,α,β)=Θ(L1)2048π6L2N71[sαβLm2bN2]3{5sαβL2+m2bN1[3β2+3α(α1)+β(2α3)]},

      ρDim3(s,α,β)=ms[2¯uu¯ss]128π4N51Θ(L1){3s2α2β2L3+m4b(α+β)N21[α(α1)+β(β1)]+2m2bsαβ×[β5+α2(α1)3+β4(5α3)+αβ(α1)2(5α2)+3β3(14α+3α2)β2(19α+17α29α3)]},

      ρDim4(s,α,β)=αsG2/π6144π4(1β)L2N51Θ(L1){s2α2β2(β1)L3[18β2+18(α1)2+β(31α36)]+m4bN21[10β6+β5(21α32)+β4(4076α+29α2+β3(24+97α95α2+37α3)+2α2(39α+11α29α3+4α4)+βα(1248α+73α259α3+26α4)+β2(654α+108α292α3+37α4)]+4sm2bαβLN1[3β(β1)4+α(β1)(3+21β32β2+16β3)+α2(β1)(932β+22β2)+α3(β1)(9+14β)+α4(β1)2α5]},

      ρDim5(s,α)=ms[3¯ugsσGu¯sgsσGs]384π4Θ(L2)(2m2b+s3sα+2sα2).

      The DimN=6, 7 and 8 terms have mixed compositions: they contain components expressed through both ρDimN(s) and ΠDimN(M2). For these terms, we find

      ΠDimN(M2,s0)=s0M2dses/M210dα1a0dβρDimN1(s,α,β)+s0M2dses/M210dαρDimN2(s,α)+10dα1a0dβΠDimN(M2,α,β).

      In the case of DimN=6, the relevant functions have the expressions

      ρDim61(s,α,β)=g3sG3m2bα510240π6(β1)LN31Θ(L1), 

      ρDim62(s,α)=Θ(L2)24π2[¯ss¯uu+g2s108π2(¯ss2+¯uu2)]×(2m2b+s3sα+2sα2),

      ΠDim6(M2,α,β)=g3sG3m4b30720M2π6α2β2L4N31exp[m2bM2N1(α+β)αβL]×{m2b(α+β)N1[5β8+2β5α2(34α)+2β3α4(54α)+3α6β(α1)+5α6(α1)2+β7(10+3α)+β4α2(5+2α(54α))β2α4(5+α(6+α))β6(5+α(3+α))]+M2αβL[11β8+8β3α4+11α6(α1)2+3βα5(α1)(5+6α)+β5α(α1)(15+8α)+2β7(11+9α)+β2α4(α1)(4+19α)+4α2β4(1+2α(α1))+β6(11+α(33+19α)]}.

      Contribution of dimension 7 is determined by the same formula (A12), where ρDim71(s,α,β), ρDim72(s,α) and ΠDim7(M2,α,β) are given by the following expressions:

      ρDim71(s,α,β)=αsG2/πms[2¯uu¯ss]768π2N31Θ(L1)αβL, ρDim72(s,α)=αsG2/πms¯uu1152π2Θ(L2)(14α+3α2),

      ΠDim7(M2,α,β) =αsG2/πm2bms[¯ss2¯uu]2304M2π2α2β2(β1)LN31exp[m2bM2N1(α+β)αβL]{2m2b(β1)(α+β)2×[β4+β3(α1)+βα2(α1)+α3(α1)+β2α(2α1)]M2αβ×[4β5+β4(α8)+4α3(α1)2+2β2(2α+α2)+β2α(13α+5α2)+βα2(19α+8α2)]}.

      The relevant functions for dimension 8 are

      ρDim81(s,α,β)=αsG2/π26144π2N31Θ(L1)αβ(α+β1),ρDim82(s,α)=¯sgsσGs¯uu48π2Θ(L2) (14α+3α2),ΠDim8(M2,α,β) =αsG2/π2m2b27648M4π2α2β2(β1)L4N31exp[m2bM2N1(α+β)αβL]{m4bα2β2(α+β)(β1)N21×[2β2+2α(α1)+β(3α2)]+M4αβL2[6β8+6α4(α1)4+3β7(5α8)+3α3β(α1)3(8α3)+β6(3654α+26α2)+β2α2(α1)2(639α+47α2)+β5(24+72α82α2+33α3)

      +β4(642α+92α299α3+47α4)+αβ3(942α+108α2133α3+58α4)]m2bM2LN1[3β5(β1)4+3αβ4(β1)3(3+5β)+α2β3(β1)2(12+β(45+38β))+2α3β2(β1)2(6+β(27+31β))+α4β(β1)(9+β(57+β(116+73β)))+α5(β1)(3+β(33+β(83+66β)))+α6(9+β(48+β(79+42β)))+α7(9+β(24+17β))+3α8(β1)]}+αsG2/π2m2b(α+β)18432π2N21exp[m2bM2N1(α+β)αβL].

      The Dim9 and Dim10 contributions are exclusively of the (A7) types

      ΠDimN(M2,s0)=10dα1a0dβΠDimN1(M2,α,β)+10dαΠDimN2(M2,α).

      For Dim9, we get

      ΠDim91(M2,α,β)=g3sG3m2bms[2¯uu¯ss]23040M6π4α4β4(β1)L4N21×R1(M2,α,β),

      and

      ΠDim92(M2,α)=αsG2/πms[¯sgsσGs3¯ugsσGu]13824M4π2α4(α1)2×R2(M2,α).

      The dimension 10 term has the following components:

      ΠDim101(M2,α,β)=αsG2/πg3sG3m2b184320M6π4α4β4(β1)L4N21×R1(M2,α,β),

      and

      ΠDim102(M2,α)=αsG2/π864M4α4(α1)2[¯ss¯uu+g2s108π2(¯ss2+¯uu2)]R2(M2,α),

      where functions R1(M2,α,β) and R2(M2,α) are given by the formulas

      R1(M2,α,β)=exp[m2bM2N1(α+β)αβL]{2M4α2β2L3[3β7+β6(α6)β4α(α1)+α4β3+3α5(α1)2+2β2α4(2α1)+β5(32α+α2)+βα(17α+6α2)]+m4b(β1)N21[5β9+5α7(α1)2+2β8(5+4α)+β5α2(5+16α16α2)+βα6(513α+8α2)]+m2bM2αβLN1[5β6(β1)3+3β5α(β1)2(5+6β)+β4α2(β1)(L+α)(16+35β)+16β2α4(β1)(12β+2β2)+βα5(β1)2(21+41β)+α6(β1)(5+β(61+60β))+3α7(β1)(7+18β)+3α8(9+10β)+11α9]},

      and

      R2(M2,α)=exp[m2bM2α(1α)][M4α3(α1)2(1+2α)4m4b(13α+3α2)+m2bM2α(827α+32α27α3)].

      In the expressions above, Θ(z) is the unit step function. We have used also the following short-hand notations:

      N1=β2+β(α1)+α(α1),  N2=(α+β)N1,  L=α+β1, L1=(1β)N21[m2bN2sαβL],    L2=sα(1α)m2b.

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