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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 baryonsΞ++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 tetraquarkT−bb;¯u¯d (hereafterT−bb ) with the massm=(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 tetraquarksbb¯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 tetraquarksQQ′¯q¯q′ may form strong-interaction stable exotic mesons, provided the ratiomQ/mq is large. Therefore, tetraquarksbb¯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 statesTQQ;¯u¯d were considered in the context of the sum rule method [18,19]. Processes in which tetraquarksTcc may be produced, namely electron-positron annihilation, heavy-ion and proton-proton collisions, andBc meson andΞbc baryon decays, also attracted the interest of researchers [20-24].The axial-vector particle
T−bb was studied in our work as well [3]. We employed the quantum chromodynamics (QCD) sum rule method and evaluated the mass of this statem=(10035±260)MeV . This means that m is below both theB−¯B∗0 andB−¯B0γ thresholds; hence, this state is a strong- and electromagnetic-interaction stable tetraquark. We also explored the semileptonic decaysT−bb →Z0bcl¯νl , whereZ0bc 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 ofT−bb 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 tetraquarksT−bb;¯u¯s andT−bb;¯u¯d (in short forms,T−b:¯s andT−b:¯d , respectively). The mass and coupling ofT−b:¯s andT−b:¯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
T−b:¯s with the same quark contentbb¯u¯s . It can be treated also as "s" member of the axial-vector multiplet of the statesbb¯u¯q . We denote this tetraquark asTAVb:¯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 massm=(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 mesonsB−B∗s andB∗−¯B0s , provided m exceeds the corresponding thresholds10695/10692MeV , respectively. The threshold for the electromagnetic decay to the final stateB−¯B0sγ is10646MeV . It is seen that even the maximal allowed value of the mass10465MeV is below all of these limits.Therefore, to evaluate the full width and lifetime of
TAVb:¯s , we analyzed the semileptonic and nonleptonic weak decaysTAVb:¯s→Z0b:¯sl¯νl andTAVb:¯s→Z0b:¯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 mesonsρ− ,K∗(892) ,D∗(2010)− , andD∗−s . The weak transitions ofTAVb:¯s can be described by the form factorsGi(q2) (i=0,1,2,3 ), which determine the differential ratesdΓ/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 andZ0b:¯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 factorsGi(q2) from the three-point sum rules for momentum transfersq2 , where this method is applicable. In that section, we also determine model functionsGi(q2) and find the partial widths of the semileptonic decaysTAVb:¯s→Z0b:¯sl¯νl . The weak nonleptonic processesTAVb:¯s→Z0b:¯sM are investigated in Section IV. This section also contains our final results for the full width and mean lifetime of the tetraquarkTAVb:¯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 tetraquarkTAVb:¯s . -
In this section, we calculate the mass
mAV and couplingfAV of the axial-vector tetraquarkTAVb:¯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 mesonZ0b:¯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 theTAVb:¯s tetraquark's decay channels. The scalar exotic mesonZ0b:¯s is a member of thebc¯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 functionΠμν(p)=i∫d4xeipx⟨0|T{Jμ(x)J†ν(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 currentJμ(x) . Among them, scalar and axial-vector diquarks are the most stable and favorable structures for composing the tetraquark state. We suggest thatTAVb:¯s is composed of the axial-vector diquarkbTCγμb and the scalar antidiquark¯uγ5C¯sT . One has to take into account that the axial-vector diquarkbTCγμb 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 theSUc(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 currentJμ(x) we use the following expressionJμ(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 functionΠ(p)=i∫d4xeipx⟨0|T{JZ(x)J†Z(0)}|0⟩.
(3) Here,
JZ(x) is the interpolating current forZ0b:¯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 andfAV . Following the standard prescriptions of the sum rule method, we expressΠμν(p) using the spectroscopic parameters ofTAVb:¯s . These manipulations generate the physical or phenomenological side of the sum rulesΠPhysμν(p) ΠPhysμν(p)=⟨0|Jμ|TAVb:¯s(p)⟩⟨TAVb:¯s(p)|J†ν|0⟩m2AV−p2+⋯.
(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 rulesΠPhysμν(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 ofJμ(x) with such a two-meson continuum generates a finite widthΓ(p2) of the tetraquark and results in the following modification [29]:1m2−p2→1m2−p2−i√p2Γ(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 few
100MeV . 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 tetraquarkTAVb:¯s is a strong-interaction stable particle, andmAV resides below the two-meson continuum, which justifies the zero-width single-pole approximation forΠPhysμν(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 stateTAVb:¯s . In terms ofmAV andfAV , the functionΠPhysμν(p) takes the formΠPhysμν(p)=m2AVf2AVm2AV−p2(−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 yieldsΠOPEμν(p)=i∫d4xeipxTr[γ5˜Sb′bs(−x)γ5Sa′au(−x)]×{Tr[γν˜Sba′b(x)γμSab′b(x)]−Tr[γν˜Saa′b(x)γμSbb′b(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 amplitudesΠPhys(p2) andΠOPE(p2) corresponding to terms∼gμν , because they are free of the scalar particles' contributions.The sum rules for
mAV andfAV 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 BorelM2 and continuum thresholds0 parameters. The second equality necessary for deriving the required sum rules is obtained by applying the operatord/d(−1/M2) to the first expression. Then, the sum rules formAV andfAV arem2AV=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 amplitudeΠOPE(p2) , andΠ′(M2,s0)= d/d(−1/M2)Π(M2,s0) . The functionΠ(M2,s0) has the following form:Π(M2,s0)=∫s0M2dsρOPE(s)e−s/M2+Π(M2),
(13) where
M=2mb+ms . The quantityρOPE(s) is the two-point spectral density, whereas the second component of the invariant amplitudeΠ(M2) includes nonperturbative contributions calculated directly fromΠOPE(p) . In the present work, we computeΠ(M2,s0) by taking into account nonperturbative terms up to the 10th dimension. The explicit expression of the functionΠ(M2,s0) is given in Appendix.The sum rules for the mass
mZ and couplingfZ of the scalar tetraquarkZ0b:¯s can be found in the same manner. The correlatorΠPhys(p) contains only a trivial Lorentz structure proportional to I, and the relevant invariant amplitude has the simple formΠPhys(p2)=m2Zf2Z/(m2Z−p2) . The QCD side of the sum rules is determined by the formulaΠOPE(p)=i∫d4xeipxTr[γ5˜Saa′b(x)γ5Sbb′c(x)]×{Tr[γ5˜Sb′bs(−x)γ5Sa′au(−x)]−Tr[γ5˜Sa′bs(−x)×γ5Sb′au(−x)]−Tr[γ5˜Sb′as(−x)γ5Sa′bu(−x)]+Tr[γ5˜Sa′ad(−x)γ5Sb′bu(−x)]}.
(14) The parameters of
Z0b:¯s after evident replacementsΠ(M2,s0)→˜Π(M2,s0) andM→˜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 functionΠOPE(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⟩ ,⟨¯qq⟩2 and⟨¯ss⟩2 ,⟨¯qq⟩⟨gsG2⟩ and⟨¯ss⟩⟨gsG2⟩ , 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⟩=m20⟨ˉss⟩,m20=(0.8±0.1)GeV2,⟨αsG2π⟩=(0.012±0.004)GeV4,⟨g3sG3⟩=(0.57±0.29)GeV6,ms=93+11−5MeV,mc=1.27±0.2GeV,mb=4.18+0.03−0.02GeV.
(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 groupSUc(3) , andA,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 ands0 . TheM2 ands0 are the auxiliary quantities, and their correct choice is one of the important problems in sum rule studies. Proper working regions forM2 ands0 must satisfy restrictions imposed on the pole contribution (PC ) and convergence of the operator product expansion measured by the ratioR(M2) , which we define respectively by the expressionsPC=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 ofM2 , the pole contribution should obeyPC>0.2 , and at the minimum ofM2 , we require fulfilment ofR(M2)⩽0.01 . Lets us note that we estimateR(M2) using the last three terms in the OPEDimN=Dim(8+9+10) .Variations of
M2 ands0 within the allowed working regions are the main sources of theoretical errors in sum rule computations. Therefore, the Borel parameterM2 should be fixed for minimizing the dependence of extracted physical quantities on its variations. The situation withs0 is more subtle, because it bears physical information about the excited states of the tetraquarkTAVb:¯s . In fact, the continuum threshold parameters0 separates the ground-state contribution from the ones of higher resonances and continuum states; hence,s0 should be below the first excitation ofTAVb:¯s . However, available information on the excited states of tetraquarks is limited to only a few theoretical studies [32-34]. As a result, one fixess0 to achieve maximalPC , ensuring fulfilment of the other constraints and simultaneously keeping the computation self-consistency under control. The latter means that the gap√s0−mAV 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 ands0 . Thus, atM2=12GeV2 , the pole contribution is0.23 , and atM2=9GeV2 , it amounts to0.62 . These values ofM2 limit the boundaries of a region in which the Borel parameter can be changed. At the minimum ofM2=9GeV2 , we getR≈0.005 . In addition, at the minimum of the Borel parameter, the perturbative contribution is 79% of the result overshooting the nonperturbative effects.For
mAV andfAV , we have obtainedmAV=(10215±250)MeV,fAV=(2.26±0.57)×10−2GeV4.
(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 couplingfAV , 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 formAV as a function ofM2 ands0 : one can see a mild dependence ofmAV on these parameters. It is also evident thatFigure 1. (color online) Dependence of the mass
mAV on the BorelM2 (left panel) and continuum thresholds0 parameters (right panel).√s0−mAV=[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 pairsB−B∗s andB∗−¯B0s , and thatTAVb:¯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 ofTAVb:¯s .The mass
mZ and couplingfZ of the stateZ0b:¯s are found from the sum rules by utilizing the following working windows forM2 ands0 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 is0.009 ; hence, the convergence of the sum rules is satisfied. The pole contributionPC atM2=6.5GeV2 andM2=5.5GeV2 equals to0.23 and0.61 , respectively. At the minimum ofM2 , the perturbative contribution constitutes 72% of the entire result and considerably exceeds that of nonperturbative terms.For
mZ andfZ , our computations yieldmZ=(6770±150)MeV,fZ=(6,3±1.3)×10−3GeV4.
(23) In Fig. 2, we depict the mass of the tetraquark
Z0b:¯s and demonstrate its dependence onM2 ands0 .Figure 2. (color online) The mass
mZ of the tetraquarkZ0b:¯s as a function of the parametersM2 (left panel) ands0 (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 dimension8 accuracy, and two lowest predictions for the mass of the axial-vector particlebb¯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 ofTAVb:¯s . As we will see below, our investigation proves thatTAVb:¯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 dimension10 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⟨¯qq⟩2 and∼⟨¯qq⟩⟨g2sG2⟩ in the light and∼⟨g3sG3⟩ in the heavy quark propagators. However, in our view, the choice of the working windows for the parametersM2 ands0 is the main source of fixed discrepancies. The regions forM2 ands0 should be extracted from the analysis of constraints (17) and (18) imposed on the invariant amplitudeΠ(M2,s0) . ThePC 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 parametersmZ andfZ approximately in the middle region of the window (22), where the pole contribution isPC≈0.42−0.45 . The working regions forM2 ands0 used in Ref. [35] ensure onlyPC≈0.31 , which may generate differences in the extracted values ofmZ . -
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 statemAV=10215MeV is480/477MeV below the thresholds10695/10692MeV for its strong decays to mesonsB−B∗s andB∗−¯B0s , respectively. The maximum of the mass10465MeV is still below these limits. The threshold10646MeV for the processTAVb:¯s→B−¯B0sγ also exceeds the maximal allowed value ofmAV , which forbids this electromagnetic decay. Therefore, the full width and mean lifetime ofTAVb:¯s are determined by its weak decays.There are different weak decay channels of
TAVb:¯s , which can be generated by sub-processesb→W−c andb→W−u . The decays triggered by the transitionb→W−c are dominant processes relative to the ones connected withb→W−u : the latter decays are suppressed relative to the dominant decays by a factor|Vbu|2/|Vbc|2 ≃0.01 , withVq1q2 being the Cabibbo-Khobayasi-Maskawa (CKM) matrix elements. In the present work, we restrict ourselves to the analysis of the dominant weak decays ofTAVb:¯s (see Fig. 3).Figure 3. (color online) The Feynman diagram for the semileptonic decay
TAVb:¯s→Z0b:¯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:¯s→Z0b:¯sl¯νl , whereas the nonleptonic transitionsTAVb:¯s→Z0b:¯sM belong to the second group. In this section, we consider the semileptonic decays and calculate the partial widths of the processesTAVb:¯s→Z0b:¯sl¯νl , where l is one of the lepton speciese,μ andτ . 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
b→W−c at the tree-level is given by the expressionHeff=GF√2Vbc¯cγμ(1−γ5)b¯lγμ(1−γ5)νl,
(24) with
GF andVbc being the Fermi coupling constant and CKM matrix element, respectively. A matrix element ofHeff between the initial and final tetraquarks is equal to⟨Z0b:¯s(p′)|Heff|TAVb:¯s(p)⟩=LμHμ,
(25) where
Lμ andHμ are the leptonic and hadronic factors, respectively. A treatment ofLμ is trivial; therefore, we consider the matrix elementHμ in a detailed form, which depends on the parameters of the tetraquarks. After factoring out the constant factors,Hμ is the matrix element of the currentJtrμ=¯cγμ(1−γ5)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 factorsGi(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)εμναβϵνpαp′β.
(27) The scaled functions
˜Gi(q2) are connected with the dimensionless form factorsGi(q2) by the equalities˜G0(q2)=˜mG0(q2), ˜Gj(q2)=Gj(q2)˜m, j=1,2,3.
(28) Here,
˜m=mAV+mZ ,pμ andϵμ are the momentum and polarization vector of the tetraquarkTAVb:¯s ,p′ is the momentum of the scalar stateZ0b:¯s . We use alsoPμ=p′μ+pμ andqμ=pμ−p′μ , the latter being the momentum transferred to the leptons. It is evident thatq2 varies withinm2l⩽q2⩽(mAV−mZ)2, whereml is the mass of a lepton l.The sum rules for the form factors
Gi(q2) can be obtainedby analyzing the three-point correlation functionΠμν(p,p′)=i2∫d4xd4yei(p′y−px)×⟨0|T{JZ(y)Jtrν(0)J†μ(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 rulesΠPhysμν(p,p′) . The functionΠPhysμν(p,p′) can be presented asΠPhysμν(p,p′)=⟨0|JZ|Z0b:¯s(p′)⟩⟨Z0b:¯s(p′)|Jtrν|TAVb:¯s(p,ϵ)⟩(p2−m2AV)(p′2−m2Z)×⟨TAVb:¯s(p,ϵ)|J†μ|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(p2−m2AV)(p′2−m2Z)×{˜G0(q2)(−gμν+pμpνm2AV)+[˜G1(q2)Pμ+˜G2(q2)qμ](−p′ν+m2AV+m2Z−q22m2AVpν)−i˜G3(q2)εμναβpαp′β}+⋯.
(32) We should also calculate the correlation function in terms of the quark propagators and find
ΠOPEμν(p,p′) . The functionΠOPEμν(p,p′) is the second side of the sum rules and has the following formΠOPEμν(p,p′)=∫d4xd4yei(p′y−px){Tr[γ5˜Sba′s(x−y)×γ5Sa′bu(x−y)](Tr[γμ˜Saa′b(y−x)γ5Sbic(y)γν(1−γ5)×Sib′b(−x)]+Tr[γμ˜Sia′b(−x)(1−γ5)γν˜Sbic(y)γ5×Sab′b(y−x)])−Tr[γ5˜Sb′as(x−y)γ5Sa′bu(x−y)]×(Tr[γμ˜Saa′b(y−x)γ5Sbic(y)γν(1−γ5)Sib′b(−x)]+Tr[γμ˜Sia′b(−x)(1−γ5)γν˜Sbic(y)γ5Sab′b(y−x)])}.
(33) To extract expressions of the form factors
Gi(q2) , we equate invariant amplitudes corresponding to the same Lorentz structures both inΠPhysμν(p,p′) andΠOPEμν(p,p′) , carry out double Borel transformations over the variablesp′2 andp2 , and perform continuum subtraction. For instance, to extract the sum rule for˜G0(q2) , we use the structuregμν , whereas for˜G3(q2) , the term∼εμναβpαp′β can be employed. The sum rules for the scaled form factors˜Gi(q2) can be written in a single formula˜Gi(M2,s0,q2)=1fAVmAVfZmZ∫s0M2dse(m2AV−s)/M21×∫s′0˜M2ds′ρi(s,s′)e(m2Z−s′)/M22,
(34) where
ρi(s,s′) are spectral densities computed as the imaginary parts of the corresponding terms inΠOPEμν(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) ands0=(s0, s′0) are the Borel and continuum threshold parameters, respectively. The pair of parameters (M21 ,s0 ) corresponds to the initial tetraquark's channels, whereas (M22 ,s′0 ) 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 tetraquarksTAVb:¯s andZ0b:¯s have been extracted in Section II. The Borel and continuum threshold parametersM2 ands0 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 ,s′0 ) 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 ratedΓ/dq2 of the semileptonic decayTAVb:¯s→Z0b:¯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 transferq2 within the limitsm2l⩽q2⩽(mAV−mZ)2 . Our results for the form factors are plotted in Fig. 4. The QCD sum rules lead to reliable predictions atm2l⩽q2⩽8GeV2 . However, these predictions do not cover the entire integration regionm2l⩽q2⩽11.87GeV2 . To solve this problem, one has to introduce extrapolation functionsGi(q2) of a relatively simple analytic form, which, atq2 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) andG1(q2) (blue squares). The solid curves are fit functionsG0(q2) andG1(q2) .For these purposes, we choose to work with the functions
Gi(q2)=Gi0exp[gi1q2m2AV+gi2(q2m2AV)2],
(35) in which the parameters
Gi0,gi1 , andgi2 should be fitted to satisfy the sum rules' predictions. The parameters of the functionsGi(q2) , obtained numerically, are presented in Table 1. The functionsGi(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=1.16637×10−5GeV−2,|Vbc|=(42.2±0.08)×10−3,
(36) me=0.511MeV ,mμ=105.658MeV , andmτ= (1776.82±0.16)MeV [38]. The predictions obtained for the partial widths of the semileptonic decaysTAVb:¯s→Z0b:¯sl¯νl are written asΓ(TAVb:¯s→Z0b:¯se−¯νe)=(5.34±1.43)×10−8MeV,Γ(TAVb:¯s→Z0b:¯sμ−¯νμ)=(5.32±1.41)×10−8MeV,Γ(TAVb:¯s→Z0b:¯sτ−¯ντ)=(2.15±0.54)×10−8MeV,
(37) and are main results of this section.
-
The second class of the weak decays of the tetraquark
TAVb:¯s are the processesTAVb:¯s→Z0b:¯sM , which may affect the full width and lifetime of the tetraquarkTAVb:¯s . Here, we study the nonleptonic weak decaysTAVb:¯s→Z0b:¯sM of the tetraquarkTAVb:¯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 mesonsT−b:¯s ,T−b:¯s ,Z0bc , andT−bs;¯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)− , andD∗−s . We provide details of analysis for the decayTAVb:¯s→Z0b:¯sρ− and write the final predictions for other channels. The relevant Feynman diagram is shown in Fig. 5.At the quark level, the effective Hamiltonian for this decay is given by the expression
Heffn.−lep=GF√2VbcV∗ud[c1(μ)Q1+c2(μ)Q2],
(38) where
Q1=(¯diui)V−A(¯cjbj)V−A,Q2=(¯diuj)V−A(¯cjbi)V−A,
(39) i and j are the color indices, and
(¯q1q2)V−A means(¯q1q2)V−A=¯q1γμ(1−γ5)q2.
(40) The short-distance Wilson coefficients
c1(μ) andc2(μ) are given on the factorization scaleμ .In the factorization method, the amplitude of the decay
TAVb:¯s→Z0b:¯sρ− has the formA=GF√2VbcV∗uda(μ)⟨ρ−(q)|(¯diui)V−A|0⟩×⟨Z0b:¯s(p′)|(¯cjbj)V−A|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)V−A|0⟩ inA can be defined in the following form⟨ρ−(q)|(¯diui)V−A|0⟩=fρmρϵ∗μ(q).
(43) Then, it is evident that
A=iGF√2fρVbcV∗uda(μ)[˜G0(q2)ϵμ(p)ϵ∗μ(q)+2˜G1(q2)(p′ϵ(p))(p′ϵ∗(q))+i˜G3(q2)εμναβϵ∗μ(q)ϵν(p)pαp′β].
(44) The decay modes
TAVb:¯s→Z0b:¯sK∗(892)[D∗(2010)− ,D∗−s] can be analyzed in a similar way. To this end, we have to replace in relevant expressions the spectroscopic parameters (mρ,fρ ) of theρ meson with masses and decay constants of the mesonsK∗(892) ,D∗(2010)− , andD∗−s and make the substitutionsVud→Vus ,Vcd , andVcs .The width of the nonleptonic decay
TAVb:¯s→Z0b:¯sρ− can be evaluated using the expressionΓ=|A|224πm2AVλ(mAV,mZ,mρ),
(45) where
λ(a,b,c)=12a[a4+b4+c4−2(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 expressionsH0=m4ρ+(m2AV−m2Z)2+2m2ρ(5m2AV−m2Z)4m2ρm2AV,H1=[m4ρ+(m2AV−m2Z)2−2m2ρ(m2AV+m2Z)]24m2ρm2AV,H2=12[m4ρ+(m2AV−m2Z)2−2m2ρ(m2AV+m2Z)],H3=−12m2ρm2AV[m6ρ+(m2AV−m2Z)3−m4ρ(m2AV+3m2Z)−m2ρ(m4AV+2m2Zm2AV−3m2Z)].
(48) In Eq. (47), we take into account that the weak form factors
˜Gj are real functions ofq2 , and their values for the processTAVb:¯s→Z0b:¯sM are fixed atq2=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 andK∗(892) are also taken from this source. The decay constants of mesonsD∗ andD∗s are theoretical predictions obtained in the lattice QCD framework [45]. The coefficientsc1(mb) , andc2(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 mD⋆s (2112.2±0.4)MeV fρ (210±4)MeV fK⋆ (204±7)MeV fD⋆ (223.5±8.4)MeV fD⋆s (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:¯s→Z0b:¯sρ− , our calculations yieldΓ(TAVb:¯s→Z0b:¯sρ−)=(3.47±0.92)×10−10MeV.
(50) The partial widths of the remaining three nonleptonic decays are presented below
Γ(TAVb:¯s→Z0b:¯sK∗(892))=(1.47±0.37)×10−11MeV,Γ(TAVb:¯s→Z0b:¯sD∗(2010)−)=(1.54±0.39)×10−11MeV,Γ(TAVb:¯s→Z0b:¯sD∗s−)=(4.97±1.32)×10−10MeV.
(51) It is evident that the parameters of the processes
TAVb:¯s→ Z0b:¯sρ− andTAVb:¯s→Z0b:¯sD∗s− are comparable to each other and may affect predictions for the tetraquarkTAVb:¯s : the other two decays can be safely neglected in the computation ofΓfull andτ . Then, using Eqs. (37), (50), and (51), we findΓfull=(12.9±2.1)×10−8MeV,τ=5.1+0.99−0.71×10−15s,
(52) which are principally new predictions of the present article.
-
We have calculated the mass, width, and lifetime of the stable axial-vector tetraquark
TAVb:¯s with the contentbb¯u¯s . This particle is a strange partner of the tetraquarkT−bb , which was explored in Ref. [3]. The width and lifetime ofT−bb ˜Γfull=(7.17±1.23)×10−8MeV,˜τ=9.18+1.90−1.34×10−15s,
(53) are comparable to those of the tetraquark
TAVb:¯s .The tetraquark
TAVb:¯s is the last of the four scalar and axial-vector statesbb¯u¯s andbb¯u¯d considered in our works. The spectroscopic parameters and widths of the scalar tetraquarksT−b:¯s andT−b:¯d were calculated in Refs. [25,26]. We demonstrated there thatT−b:¯s andT−b:¯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 diquarkbb and light antidiquarks is presented in Table 3.Tetraquark (JP) Mass/ MeV Width/ MeV Lifetime TAVb:¯s(1+) 10215±250 (12.9±2.1)×10−8 5.1+0.99−0.71 fs T−bb(1+) 10035±260 (7.17±1.23)×10−8 9.18+1.90−1.34 fs T−b:¯s(0+) 10250±270 (15.21±2.59)×10−10 0.433+0.089−0.063 ps T−b:¯d(0+) 10135±240 (10.80±1.88)×10−10 0.605+0.126−0.089 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 to35MeV , and for particles with quark contentbb¯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
2−3 orders of magnitude. The widths of the scalar tetraquarksT−b:¯s andT−b:¯d are considerably smaller than the widths of the axial-vector particlesT−bb andTAVb:¯s . As a result, the mean lifetimes of the scalar tetraquarks are ~1ps , whereas for the axial vector states, we getτ≈10 fs . Stated differently, the scalar tetraquarksT−b:¯s andT−b:¯d are heavier and live longer than the corresponding axial-vector particles.The spectroscopic parameters and lifetimes of the axial-vector states
T−bb andTAVb:¯s were also explored in Refs. [1,5]. The lifetime367fs of the stateT−bb predicted in Ref. [1] is considerably longer than our result9.18fs. The lifetimesτ≃800fs of the tetraquarksT−bb andTAVb:¯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 ofTAVb:¯s using Eqs. (50) and (51) and foundτ≃753fs . Despite the fact that the channels that were explored in Ref. [5] differ from the decays that were considered in the present work, forτ , 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 ofTAVb:¯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. -
In the present work, we use the light quark propagator
Sabq(x) , which is given by the following formulaSabq(x)=iδab⧸x2π2x4−δabmq4π2x2−δab⟨¯qq⟩12+iδab⧸xmq⟨¯qq⟩48−δabx2192⟨¯qgsσGq⟩+iδabx2⧸xmq1152⟨¯qgsσGq⟩−igsGαβab32π2x2[⧸xσαβ+σαβ⧸x]−iδabx2⧸xg2s⟨¯qq⟩27776−δabx4⟨¯qq⟩⟨g2sG2⟩27648+⋯.
For the heavy quarks Q , we utilize the propagator
SabQ(x) SabQ(x)=i∫d4k(2π)4e−ikx{δab(⧸k+mQ)k2−m2Q−gsGαβab4σαβ(⧸k+mQ)+(⧸k+mQ)σαβ(k2−m2Q)2+g2sG212δabmQk2+mQ⧸k(k2−m2Q)4+g3sG348δab(⧸k+mQ)(k2−m2Q)6[⧸k(k2−3m2Q)+2mQ(2k2−m2Q)](⧸k+mQ)+⋯}.
Above, we have used the notation
Gαβab≡GαβAtAab,
where
GαβA is the gluon field strength tensor, andtA=λA/2 withλA being the Gell-Mann matrices,A=1,2,⋯,8. The invariant amplitude
ΠOPE(p2) used for calculating the mass and coupling of the tetraquarkT−b:¯s after the Borel transformation and subtraction procedures takes the following formΠ(M2,s0)=∫s0M2dsρOPE(s)e−s/M2+Π(M2),
where
ρOPE(s)=ρpert.(s)+8∑N=3ρDimN(s), Π(M2)=10∑N=6ΠDimN(M2).
Components of the spectral density are given by the formulas
ρ(s)=∫10dα∫1−a0dβρ(s,α,β), ρ(s)=∫10dαρ(s,α),
depending on whether
ρ(s,α,β) is a function ofα andβ or onlyα . The same is true also for termsΠ(M2) , i.e.,ΠDimN(M2)=∫10dα∫1−a0dβΠDimN(M2,α,β),Π(M2)=∫10dαΠDimN(M2,α).
In these expressions,
α andβ 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[sαβL−m2bN2]3{5sαβL2+m2bN1[3β2+3α(α−1)+β(2α−3)]},
ρDim3(s,α,β)=ms[2⟨¯uu⟩−⟨¯ss⟩]128π4N51Θ(L1){−3s2α2β2L3+m4b(α+β)N21[α(α−1)+β(β−1)]+2m2bsαβ×[β5+α2(α−1)3+β4(5α−3)+αβ(α−1)2(5α−2)+3β3(1−4α+3α2)−β2(1−9α+17α2−9α3)]},
ρDim4(s,α,β)=⟨αsG2/π⟩6144π4(1−β)L2N51Θ(L1){−s2α2β2(β−1)L3[18β2+18(α−1)2+β(31α−36)]+m4bN21[10β6+β5(21α−32)+β4(40−76α+29α2+β3(−24+97α−95α2+37α3)+2α2(3−9α+11α2−9α3+4α4)+βα(12−48α+73α2−59α3+26α4)+β2(6−54α+108α2−92α3+37α4)]+4sm2bαβLN1[3β(β−1)4+α(β−1)(−3+21β−32β2+16β3)+α2(β−1)(9−32β+22β2)+α3(β−1)(−9+14β)+α4(β−1)−2α5]},
ρDim5(s,α)=ms[3⟨¯ugsσGu⟩−⟨¯sgsσGs⟩]384π4Θ(L2)(2m2b+s−3sα+2sα2).
The
DimN=6 ,7 and8 terms have mixed compositions: they contain components expressed through bothρDimN(s) andΠDimN(M2) . For these terms, we findΠDimN(M2,s0)=∫s0M2dse−s/M2∫10dα∫1−a0dβρDimN1(s,α,β)+∫s0M2dse−s/M2∫10dαρDimN2(s,α)+∫10dα∫1−a0dβΠDimN(M2,α,β).
In the case of
DimN=6 , the relevant functions have the expressionsρDim61(s,α,β)=−⟨g3sG3⟩m2bα510240π6(β−1)LN31Θ(L1),
ρDim62(s,α)=Θ(L2)24π2[⟨¯ss⟩⟨¯uu⟩+g2s108π2(⟨¯ss⟩2+⟨¯uu⟩2)]×(2m2b+s−3sα+2sα2),
ΠDim6(M2,α,β)=−⟨g3sG3⟩m4b30720M2π6α2β2L4N31exp[−m2bM2N1(α+β)αβL]×{m2b(α+β)N1[5β8+2β5α2(3−4α)+2β3α4(5−4α)+3α6β(α−1)+5α6(α−1)2+β7(−10+3α)+β4α2(−5+2α(5−4α))−β2α4(5+α(−6+α))−β6(−5+α(3+α))]+M2αβL[11β8+8β3α4+11α6(α−1)2+3βα5(α−1)(−5+6α)+β5α(α−1)(−15+8α)+2β7(−11+9α)+β2α4(α−1)(−4+19α)+4α2β4(1+2α(α−1))+β6(11+α(−33+19α)]}.
Contribution of dimension 7 is determined by the same formula (A12), where
ρDim71(s,α,β) ,ρDim72(s,α) andΠDim7(M2,α,β) are given by the following expressions:ρDim71(s,α,β)=⟨αsG2/π⟩ms[2⟨¯uu⟩−⟨¯ss⟩]768π2N31Θ(L1)αβL, ρDim72(s,α)=−⟨αsG2/π⟩ms⟨¯uu⟩1152π2Θ(L2)(1−4α+3α2),
ΠDim7(M2,α,β) =⟨αsG2/π⟩m2bms[⟨¯ss⟩−2⟨¯uu⟩]2304M2π2α2β2(β−1)LN31exp[−m2bM2N1(α+β)αβL]{2m2b(β−1)(α+β)2×[β4+β3(α−1)+βα2(α−1)+α3(α−1)+β2α(2α−1)]−M2αβ×[4β5+β4(α−8)+4α3(α−1)2+2β2(2−α+α2)+β2α(1−3α+5α2)+βα2(1−9α+8α2)]}.
The relevant functions for dimension 8 are
ρDim81(s,α,β)=−⟨αsG2/π⟩26144π2N31Θ(L1)αβ(α+β−1),ρDim82(s,α)=−⟨¯sgsσGs⟩⟨¯uu⟩48π2Θ(L2) (1−4α+3α2),ΠDim8(M2,α,β) =−⟨αsG2/π⟩2m2b27648M4π2α2β2(β−1)L4N31exp[−m2bM2N1(α+β)αβL]{m4bα2β2(α+β)(β−1)N21×[2β2+2α(α−1)+β(3α−2)]+M4αβL2[6β8+6α4(α−1)4+3β7(5α−8)+3α3β(α−1)3(8α−3)+β6(36−54α+26α2)+β2α2(α−1)2(6−39α+47α2)+β5(−24+72α−82α2+33α3)
+β4(6−42α+92α2−99α3+47α4)+αβ3(9−42α+108α2−133α3+58α4)]−m2bM2LN1[3β5(β−1)4+3αβ4(β−1)3(−3+5β)+α2β3(β−1)2(12+β(−45+38β))+2α3β2(β−1)2(6+β(−27+31β))+α4β(β−1)(−9+β(−57+β(−116+73β)))+α5(β−1)(−3+β(33+β(−83+66β)))+α6(−9+β(48+β(−79+42β)))+α7(9+β(−24+17β))+3α8(β−1)]}+⟨αsG2/π⟩2m2b(α+β)18432π2N21exp[−m2bM2N1(α+β)αβL].
The
Dim9 andDim10 contributions are exclusively of the (A7) typesΠDimN(M2,s0)=∫10dα∫1−a0dβΠDimN1(M2,α,β)+∫10dαΠDimN2(M2,α).
For
Dim9 , we getΠDim91(M2,α,β)=⟨g3sG3⟩m2bms[2⟨¯uu⟩−⟨¯ss⟩]23040M6π4α4β4(β−1)L4N21×R1(M2,α,β),
and
ΠDim92(M2,α)=⟨αsG2/π⟩ms[⟨¯sgsσGs⟩−3⟨¯ugsσGu⟩]13824M4π2α4(α−1)2×R2(M2,α).
The dimension 10 term has the following components:
ΠDim101(M2,α,β)=−⟨αsG2/π⟩⟨g3sG3⟩m2b184320M6π4α4β4(β−1)L4N21×R1(M2,α,β),
and
ΠDim102(M2,α)=−⟨αsG2/π⟩864M4α4(α−1)2[⟨¯ss⟩⟨¯uu⟩+g2s108π2(⟨¯ss⟩2+⟨¯uu⟩2)]R2(M2,α),
where functions
R1(M2,α,β) andR2(M2,α) are given by the formulasR1(M2,α,β)=exp[−m2bM2N1(α+β)αβL]{−2M4α2β2L3[3β7+β6(α−6)−β4α(α−1)+α4β3+3α5(α−1)2+2β2α4(2α−1)+β5(3−2α+α2)+βα(1−7α+6α2)]+m4b(β−1)N21[5β9+5α7(α−1)2+2β8(−5+4α)+β5α2(−5+16α−16α2)+βα6(5−13α+8α2)]+m2bM2αβLN1[5β6(β−1)3+3β5α(β−1)2(−5+6β)+β4α2(β−1)(L+α)(−16+35β)+16β2α4(β−1)(1−2β+2β2)+βα5(β−1)2(−21+41β)+α6(β−1)(5+β(−61+60β))+3α7(β−1)(−7+18β)+3α8(−9+10β)+11α9]},
and
R2(M2,α)=exp[−m2bM2α(1−α)][M4α3(α−1)2(1+2α)−4m4b(1−3α+3α2)+m2bM2α(8−27α+32α2−7α3)].
In the expressions above,
Θ(z) is the unit step function. We have used also the following short-hand notations:N1=β2+β(α−1)+α(α−1), N2=(α+β)N1, L=α+β−1, L1=(1−β)N21[m2bN2−sαβL], L2=sα(1−α)−m2b.
A family of double-beauty tetraquarks: Axial-vector state {{T}^{{-}}_{{bb};\overline{{u}}\overline{{s}}}}}
- Received Date: 2020-07-13
- Available Online: 2021-01-15
Abstract: The spectroscopic parameters and decay channels of the axial-vector tetraquark