Spectroscopic parameters and decays of the resonance \(Z_b(10610)\)

The European Physical Journal C, Dec 2017

The resonance \(Z_b(10610)\) is investigated as the diquark–antidiquark \( Z_b=[bu][\overline{bd}]\) state with spin–parity \(J^{P}=1^{+}\). The mass and current coupling of the resonance \(Z_b(10610)\) are evaluated using QCD two-point sum rule and taking into account the vacuum condensates up to ten dimensions. We study the vertices \(Z_b\Upsilon (nS)\pi \ (n=1,2,3)\) by applying the QCD light-cone sum rule to compute the corresponding strong couplings \(g_{Z_b\Upsilon (nS)\pi }\) and widths of the decays \(Z_b \rightarrow \Upsilon (nS)\pi \). We explore also the vertices \(Z_b h_{b}(mP)\pi \ (m=1,2)\) and calculate the couplings \(g_{Z_b h_{b}(mP)\pi }\) and the widths of the decay channels \(Z_b \rightarrow h_{b}(mP)\pi \). To this end, we calculate the mass and decay constants of the \(h_b(1P)\) and \(h_b(2P) \) mesons. The results obtained are compared with experimental data of the Belle Collaboration.

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Spectroscopic parameters and decays of the resonance \(Z_b(10610)\)

Eur. Phys. J. C Spectroscopic parameters and decays of the resonance Zb(10610) S. S. Agaev 2 K. Azizi 0 1 H. Sundu 3 0 School of Physics, Institute for Research in Fundamental Sciences (IPM) , P. O. Box 19395-5531, Tehran , Iran 1 Department of Physics, Dogˇus ̧ University , Acibadem-Kadiköy, 34722 Istanbul , Turkey 2 Institute for Physical Problems, Baku State University , 1148 Baku , Azerbaijan 3 Department of Physics, Kocaeli University , 41380 Izmit , Turkey The resonance Zb(10610) is investigated as the diquark-antidiquark Zb = [bu][bd] state with spin-parity J P = 1+. The mass and current coupling of the resonance Zb(10610) are evaluated using QCD two-point sum rule and taking into account the vacuum condensates up to ten dimensions. We study the vertices Zbϒ (n S)π (n = 1, 2, 3) by applying the QCD light-cone sum rule to compute the corresponding strong couplings gZbϒ(nS)π and widths of the decays Zb → ϒ (n S)π . We explore also the vertices Zbhb(m P)π (m = 1, 2) and calculate the couplings gZbhb(m P)π and the widths of the decay channels Zb → hb(m P)π . To this end, we calculate the mass and decay constants of the hb(1 P) and hb(2 P) mesons. The results obtained are compared with experimental data of the Belle Collaboration. - 1 Introduction Discovery of the charged resonances which cannot be explained as c¯c or b¯b states has opened a new page in the physics of exotic multi-quark systems. The first tetraquarks of this family are Z ±(4430) states which were observed by the Belle Collaboration in B meson decays B → K ψ π ± as resonances in the ψ π ± invariant mass distributions [ 1 ]. The masses and widths of these states were repeatedly measured and refined. Recently, the LHCb Collaboration confirmed the existence of the Z −(4430) structure in the decay B0 → K +ψ π − and unambiguously determined that its spin–parity is J P = 1+ [ 2,3 ]. They also measured the mass and width of Z −(4430) resonance and updated the existing experimental data. Two charmoniumlike resonances, Z1(4050) and Z2(4250), were discovered by the Belle Collaboration in the decay B¯ 0 → K −π +χc1, emerging as broad peaks in the χc1π invariant mass distribution [4]. Famous members of the charged tetraquark family Zc± (3900) were observed by the BESIII Collaboration in the process e+e− → J /ψ π +π − as resonances with J P = 1+ in the J /ψ π ± mass distribution [ 5 ]. The charged state Zc(4020) was also found by the BESIII Collaboration in two different processes, e+e− → hcπ +π − and e+e− → (D D¯ )±π ∓ (see Refs. [ 6,7 ]). There is another charged state, namely the Zc(4200) resonance which was detected and announced by Belle [ 8 ]. All aforementioned resonances belong to the class of the charmonium-like tetraquarks, and contain a c¯c pair and light quarks (antiquarks). They were mainly interpreted as diquark–antidiquark systems or bound states of D and/or D mesons. It is remarkable that the b-counterparts of the charmoniumlike states, i.e. charged resonances composed of a b¯b pair and light quarks were found as well. Thus, the Belle Collaboration discovered the resonances Zb(10610) and Zb(10650) (hereafter, Zb and Zb, respectively) in the decays ϒ (5S) → ϒ (n S)π +π −, n = 1, 2, 3 and ϒ (5S) → hb(m P)π +π −, m = 1, 2 [ 9,10 ]. These two states with favored spin–parity J P = 1+ appear as resonances in the ϒ (n S)π ± and hb(m P)π ± mass distributions. The masses of the Zb and Zb resonances are m = (10607.2 ± 2.0) MeV, m = (10652.2 ± 1.5) MeV, respectively. The width of the Zb state averaged over five decay channels equals = (18.4 ± 2.4) MeV, whereas the average width of Zb is = (11.5 ± 2.2) MeV. Recently, the dominant decay channel of Zb, namely the Zb → B+ B¯ 0 + B¯ 0 B + process, was also observed [11]. In this work fractions of different channels of Zb and Zb resonances were reported as well. Further information on the (1) experimental status of the Zb and Zb states and other heavy exotic mesons and baryons can be found in Ref. [ 12 ]. The existence of hidden-bottom states, i.e. of the Zb resonances, was foreseen before their experimental observation. Thus, in Ref. [ 13 ] the authors suggested to look for the diquark–antidiquark systems with bb¯ud¯ content as peaks in the invariant mass of the ϒ (1S)π and ϒ (2S)π systems. The existence of the molecular state B B¯ was predicted in Ref. [ 14 ]. After discovery of the Zb resonances theoretical studies of the charged hidden-bottom states became more intensive and fruitful. In fact, articles devoted to the structures and decay channels of the Zb states encompass all existing models and computational schemes suitable to study the multiquark systems. Thus, in Refs. [ 15,16 ] the spectroscopic and decay properties of Zb and Zb were explored using the heavy quark symmetry by modeling them as J = 1 S-wave molecular B B¯ –B B¯ states and B B¯ , respectively. The existence of similar states with quantum numbers 0+, 1+, 2+ was predicted as well. The diquark–antidiquark interpretation of the Zb states was proposed in Refs. [ 17,18 ]. It was demonstrated that Belle results on the decays ϒ (5S) → ϒ (n S)π +π − and ϒ (5S) → hb(m P)π +π − support Zb resonances as diquark–antidiquark states. This analysis is based on a scheme for the spin–spin quark interactions inside diquarks originally suggested and successfully used to explore hiddencharm resonances [ 19 ]. The Zb resonance was considered in Ref. [ 20 ] as a B B¯ molecular state, where its mass was computed in the context of the QCD sum rule method. The prediction for the mass m B B¯ = 10.54 ± 0.22 GeV obtained there allowed the authors to conclude that Zb could be a B B¯ molecular state. Similar conclusions were also drawn in the framework of the chiral quark model. Indeed, in Ref. [ 21 ] the B B¯ and B B¯ bound states with J PC = 1+− were studied in the chiral quark model, and found to be good candidates for the Zb and Zb resonances. Moreover, the existence of molecular states B B¯ with J PC = 1++, and B B¯ with J PC = 0++, 2++ was predicted. Explorations performed using the one bosonexchange model also led to the molecular interpretations of the Zb and Zb resonances [ 22 ]. However, an analysis carried out in the framework of the Bethe–Salpeter approach demonstrated that two heavy mesons can form an isospin singlet bound state but cannot form an isotriplet compound. Hence, the Zb resonance presumably is a diquark–antidiquark, but not a molecular state [ 23 ]. Both the diquark–antidiquark and the molecular pictures for the internal organization of Zb and Zb within the QCD sum rules method were examined in Ref. [ 24 ]. In this work the authors constructed different interpolating currents with I G J P = 1+1+ to explore the Zb and Zb states and evaluate their masses. Among alternative interpretations of the Zb states it is worth to note Refs. [ 25,26 ], where the peaks observed by the Belle Collaboration were explained as cusp and coupling channel effects, respectively. Theoretical studies that address problems of the Zb states are numerous (see Refs. [ 27–36 ]). An analysis of these and other investigations can be found in Refs. [ 37,38 ]. As is seen, the theoretical status of the resonances Zb and Zb remains controversial and deserves further and detailed explorations. In the present work we are going to calculate the spectroscopic parameters of Zb = [bu][bd] state by assuming that it is a tetraquark state with a diquark–antidiquark structure and positive charge. We use QCD two-point sum rules to evaluate its mass and current coupling by taking into account vacuum condensates up to ten dimensions. We also investigate five observed decay channels of the Zb resonance employing QCD sum rules on the light-cone. As a byproduct, we derive the mass and decay constant of the hb(m P), m = 1, 2 mesons. This work has the following structure: In Sect. 2 we calculate the mass and current coupling of the Zb resonance. In Sect. 3 we analyze the decay channels Zb → ϒ (n S)π, n = 1, 2, 3, and we calculate their widths. Section 4 is devoted to an investigation of the decay modes Zb → hb(m P)π, m = 1, 2, and it consists of two subsections. In the first subsection we calculate the mass and decay constant of the hb(1 P) and hb(2 P) mesons. To this end, we employ the two-point sum rule approach by including into the analysis condensates up to eight dimensions. In the next subsection using the parameters of the hb(m P) mesons we evaluate the widths of the decays under investigation. The last section is reserved for an analysis of the obtained results and a discussion of possible interpretations of the Zb resonance. 2 Mass and current coupling of the Zb state: QCD two-point sum rule predictions In this section we derive QCD sum rules to calculate the mass and current coupling of the Zb state by suggesting that it has a diquark–antidiquark structure with quantum numbers I G J P = 1+1+. To this end, we begin from the two-point correlation function μν ( p) = i d4x ei px 0|T { JμZb (x ) JνZb†(0)}|0 , (2) where JμZb (x ) is the interpolating current for the Zb state with the required quark content and quantum numbers. It is possible to construct various currents to interpolate the Zb and Zb resonances. One of them is a [ub]S=0[db]S=1– [ub]S=1[db]S=0 type diquark–antidiquark current that is used to consider the Zb state, uaT (x )C γ5bb(x ) dd (x )γμC beT (x ) − uaT (x )C γμbb(x ) dd (x )γ5C beT (x ) . (3) The current for Zb can be defined in the form Z Jμ b (x ) = √˜ εμναβ uaT (x )C γ ν bb(x ) 2 ×Dα dd (x )γ β C beT (x ) , where Dα = ∂α − i gs Aα(x ) [ 24 ]. In Eqs. (3) and (4) we have introduced the notation = abc and ˜ = dec. In above expressions a, b, c, d and e are color indices, and C is the charge conjugation matrix. By choosing different currents to interpolate the Zb and Zb resonances one treats both of them as ground-state particles in the corresponding sum rules. We also follow this approach and use the current JμZb (x ) to calculate the mass and current coupling of the Zb state. To find the QCD sum rules we first have to calculate the correlation function in terms of the physical degrees of freedom. To this end, we saturate μν ( p) with a complete set of states with the quantum numbers of the Zb resonance and perform in Eq. (2) an integration over x to get Pμhνys( p) = 0| JμZb |Zb( p) Zb( p)| JνZb†|0 m2Zb − p2 where m Zb is the mass of the Zb state, and dots indicate the contributions of higher resonances and continuum states. We define the current coupling f Zb through the matrix element 0| JμZb |Zb( p) = f Zb m Zb εμ, with εμ being the polarization vector of Zb state. Then in terms of m Zb and f Zb , the correlation function can be written in the following form: m2Zb f Z2b Pμhνys( p) = m2Zb − p2 −gμν + (9) (10) (11) interpolating current JμZb and quark propagators. After contracting in Eq. (2) the b-quark and light quark fields we get ×γ5 Sbbb (x ) Tr γμ Sbe e(−x )γν Sdd d (−x ) −Tr γμ Sbe e(−x )γ5 Sdd d (−x ) Tr γν Suaa (x ) × γ5 Sbbb (x ) − Tr γ5 Sua a(x )γμ Sbb b(x ) ×Tr γ5 Sbe e(−x )γν Sdd d (−x ) + Tr γν Suaa (x ) × γμ Sbbb (x ) Tr γ5 Sbe e(−x )γ5 Sdd d (−x ) , where Sb(q)(x ) = C Sbij(Tq)(x )C. i j In the expressions above Sqab(x ) and Sbab(x ) are the light u-, and d-, and the heavy b-quark propagators, respectively. We choose the light quark propagator Sqab(x ) in the form x/ mq qq Sqab(x ) = i δab 2π 2x 4 − δab 4π 2x 2 − δab 12 +i δab x/mq qq 48 x 2 − δab 192 q gsσ Gq x 2x/mq q gsσ Gq +i δab 1152 αβ i gsGab − 32π 2x 2 x/σαβ + σαβ x/ − i δab −δab x 4 qq gs2G2 27648 For the b-quark propagator Sbab(x ) we employ the expression Sbab(x) = i d4k (2π)4 e−ikx δab (k/ + mb) k2 − mb2 − gs Gaαβb σαβ (k/ + mb) + (k/ + mb) σαβ 4 (k2 − m2b)2 + gs21G22 δabmb (k2 − mb2)4 + k2 + mbk/ gs3G3 (k/ + mb) 48 δab (k2 − mb2)6 × k/ k2 − 3m2b + 2mb 2k2 − mb2 (k/ + mb) + . . . . pμ pν m2Zb At the next stage we derive the theoretical expression for the correlation function μQνCD( p) in terms of the quark– gluon degrees of freedom. It can be determined using the where A, B, C = 1, 2 . . . 8. In Eq. (11) t A = λA/2, λA are the Gell-Mann matrices, and the gluon field strength tensor GαAβ ≡ GαAβ (0) is fixed at x = 0. The QCD sum rule can be obtained by choosing the same Lorentz structures in both Pμhνys( p) and μQνCD( p). We work with terms ∼ gμν , which do not contain the effects of spin-0 particles. The invariant amplitude QCD( p2) corresponding to this structure can be written down as the dispersion integral QCD( p2) = ∞ ρQCD(s) 4m2b s − p2 ds + . . . , where ρQCD(s) is the corresponding spectral density. It is a key ingredient of sum rules for m2Zb and f Z2b and can be obtained using the imaginary part of the invariant amplitude QCD( p2). Methods of such calculations are well known and presented numerously in the literature. Therefore, we omit further details, emphasizing only that ρQCD(s) in the present work is calculated by including into the analysis quark, gluon and mixed condensates up to ten dimensions. After applying the Borel transformation on the variable p2 to QCD( p2), equating the expression obtained to B Phys( p), and subtracting the continuum contribution, we obtain the required sum rules. Thus, the mass of the Zb state can be evaluated from the sum rule m2Zb = 4sm02b dssρQCD(s)e−s/M2 4sm02b dsρQCD(s)e−s/M2 , whereas for the current coupling f Zb we employ the formula 1 f 2 Zb = m2Zb 4m2b s0 dsρQCD(s)e(m2Zb −s)/M2 . The sum rules for m Zb and f Zb depend on different vacuum condensates stemming from the quark propagators, on the mass of the b-quark, and on the Borel variable M 2 and continuum threshold s0, which are auxiliary parameters of the numerical computations. The vacuum condensates are (12) (13) (14) parameters that do not depend on the problem under consideration: their numerical values extracted once from some processes are applicable in all sum rule computations. For quark and mixed condensates in the present work we employ q¯q = −(0.24 ± 0.01)3 GeV3, q gsσ Gq = m20 q¯q , where m2 0 = (0.8 ± 0.1) GeV2, whereas for the gluon condensates we utilize αsG2/π = (0.012 ± 0.004) GeV4, gs3G3 = (0.57 ± 0.29) GeV6. The mass of the b−quark can be found in Ref. [ 46 ]: it is equal to mb = 4.18+−00..0043 GeV. The choice of the Borel parameter M 2 and continuum threshold s0 should obey some restrictions of sum rule calculations. Thus, the limits within of which M 2 can be varied (working window) are determined from convergence of the operator product expansion and dominance of the pole contribution. In the working window of the threshold parameter s0 the dependence of the quantities on M 2 should be minimal. In real calculations, however, the quantities of interest depend on the parameters M 2 and s0, which affects the accuracy of the extracted numerical values. Theoretical errors in sum rule calculations may amount to 30% of the predictions obtained, and a considerable part of these ambiguities are connected namely with the choice of M 2 and s0. An analysis performed in accordance with these requirements allows us to fix the working windows for M 2 and s0: M 2 = 9 − 12 GeV2, s0 = 123 − 127 GeV2. (15) In Figs. 1 and 2 we demonstrate the results of the numerical computations of the mass m Zb and the current coupling f Zb as functions of the parameters M 2 and s0. As is seen, m Zb and f Zb are rather stable within the working windows of the auxiliary parameters, but there is still a dependence on them in the plotted figures. Our results for m Zb and f Zb read m Zb = 10581+−114624 MeV, f Zb = (2.79+−00..5655) × 10−2 GeV4. (16) 9.5 10.0 10.5 11.0 11.5 12.0 124 126 127 10.0 9.0 11.0 10.8 V10.6 e G Zb10.4 m 10.2 10.0 123 Within theoretical errors m Zb is in agreement with experimental measurements of the Belle Collaboration (1). The mass and current coupling Zb given by Eq. (16) will be used as input parameters in the next sections to find the widths of the decays Zb → ϒ (n S)π and Zb → hb(m P)π . 3 Decay channels Zb → ϒ(nS)π, n = 1, 2, 3. This section is devoted to the calculation of the width of the Zb → ϒ (n S)π, n = 1, 2, 3, decays. To this end we determine the strong couplings gZbϒnπ , n = 1, 2, 3 (in the formulas we utilize ϒn ≡ ϒ (n S)) using QCD sum rules on the light-cone in conjunction with the ideas of the soft-meson approximation. We start from an analysis of the vertices Zbϒnπ aiming to calculate gZbϒnπ , and therefore we consider the correlation function μν ( p, q) = i d4x ei px π(q) T { Jμϒ (x ) JνZb†(0)} 0 , where Jμϒ (x ) = bi (x )γμbi (x ) is the interpolating current for mesons ϒ (n S). Here p, q and p = p + q are the momenta of ϒ (n S), π and Zb, respectively. To derive the sum rules for the couplings gZbϒnπ , we calculate μν ( p, q) in terms of the physical degrees of freedom. It is not difficult to obtain (17) (18) Pμhνys( p, q) = 3 n=1 × 0 J ϒ ϒn ( p) μ p2 − m2ϒ(nS) Zb( p )| JνZb†|0 For calculation of the strong couplings we choose to work with the structure ∼ gμν . To this end, we have to isolate the invariant function Phys( p2, p 2) corresponding to this structure and find its double Borel transformation. But it is well known that in the case of vertices involving a tetraquark and two conventional mesons one has to set q = 0 [ 39 ]. This is connected with the fact that the interpolating current for the tetraquark is composed of four quark fields and, after contracting two of them in the correlation function μν ( p, q) with the relevant quark fields from the heavy meson’s current, we encounter a situation where the remaining quarks are located at the same space-time point. These quarks fields, sandwiched between a light meson and vacuum instead of generating light meson’s distribution amplitudes, create the local matrix elements. Then, in accordance with the fourmomentum conservation at such vertices, we have to set q = 0. In QCD light-cone sum rules the limit q → 0, when the light-cone expansion reduces to a short-distant expansion over local matrix elements, is known as the “soft-meson approximation”. The mathematical methods to handle the soft-meson limit were elaborated in Refs. [ 40,41 ] and were successfully applied to tetraquark vertices in our previous articles [ 42–45 ]. In the soft limit p = p and the relevant invariant amplitudes in the correlation function depend only on one variable: p2. In the present work we use this approach, which implies calculation of the correlation function with equal initial and final momenta, p = p, and dealing with the double pole terms obtained. In fact, in the limit p = p we replace in Eq. (21) 1 , where α and β are the spinor indices. We continue and use the expansion ua dd 1 j α β → 4 βα u a j dd , where m2n = (m2Zb + m2ϒn )/2, and we carry out the Borel transformation over p2. Then for the Borel transformation of Phys( p2) we get B Phys( p2) = Now one has to derive the correlation function in terms of the quark–gluon degrees of freedom and find the QCD side of the sum rules. Contracting of the heavy quark fields in Eq. (17) yields μQνCD( p, q) = d4x ei px √2 γ5 Sbib(x )γμ ×Sbei (−x )γν + γν Sbib(x )γμ Sbei (−x )γ5 × π(q)|uaα(0)dβd (0)|0 , (22) αβ (23) (24) where j is the full set of Dirac matrices j = 1, γ5, γλ, i γ5γλ, σλρ /√2. Replacing ua dβd in Eq. (23 ) by this expansion and performα ing summations over color indices it is not difficult to determine local matrix elements of the pion which contribute to μQνCD( p, q) (see Ref. [ 39 ] for details). It turns out that in the soft limit the pion’s local matrix element which contributes to Im μQνCD( p, q = 0) is (25) (26) (27) (28) 0|d(0)i γ5u(0)|π(q) = fπ μπ , where m2π μπ = mu + md . After fixing in Im μQνCD( p, q = 0) the structure ∼ gμν it is straightforward to extract ρϒQCD(s) as a sum of the perturbative and nonperturbative components: fπ μπ ρQCD(s) = 12√2 ϒ ρpert.(s) + ρn.−pert.(s) . The ρQCD(s) can be obtained after the replacement mc → ϒ mb from the spectral density of Zc → J /ψ π decay calculated in Refs. [ 39,45 ]. Its perturbative component ρpert.(s) has a simple form and reads The nonperturbative contribution ρn.−pert.(s) depends on the vacuum expectation values of the gluon operators and contains terms of four, six and eight dimensions. Its explicit expression was presented in the appendix of Ref. [ 45 ]. The continuum subtraction in the case under consideration can be done using quark–hadron duality, which leads to the desired sum rule for the strong couplings. We get 3 n=1 = s0 4m2b gZbϒnπ fϒn f Zb m Zb mϒn mn2 e−mn2/M2 M 2 dse−s/M2 ρQCD(s). ϒ Here some comments are in order on the expression obtained, Eq. (28). It is well known that the soft limit considerably simplifies the QCD side of light-cone sum rule expressions [ 40 ]. At the same time, in the limit q → 0 the phenomenological side of the sum rules gains contributions which are not suppressed relative to a main term. In our case the main term corresponds to vertex Zbϒ (1S)π , where the tetraquark and mesons are ground-state particles. Additional contributions emerge due to vertices Zbϒ π where some of particles (or all of them) are on their excited states. In Eq. (28) terms corresponding to vertices Zbϒ (2S)π and Zbϒ (3S)π belong to this class of contributions. When we are interested in extraction of parameters of a vertex built of only ground-state particles, these additional contributions are undesired contaminations which may affect the accuracy of the calculations. A technique to eliminate them from sum rules is also well known [ 40,41 ]. To this end, in accordance with elaborated recipes one has to act by the operator P(M 2, mn2) = 1 − M 2 d d M 2 M 2emn2/M2 (29) to Eq. (28). In the present work we are going to evaluate three strong couplings gZbϒnπ and therefore we use the original form of the sum rule given by Eq. (28). But it provides only one equality for three unknown quantities. In order to get two additional equations we act by operators d/d(−1/M 2) and d2/d(−1/M 2)2 to both sides of Eq. (28) and solve obtained equations to find gZbϒnπ . The widths of the decays Zb → ϒ (n S)π, n = 1, 2, 3 can be calculated applying the standard methods and have the same form as in the case of the decay Zc → J /ψ π . After evident replacements in the corresponding formula we get (Zb → ϒnπ ) = g2 m2 Zbϒnπ ϒn λ m Zb , mϒn , mπ 24π × 3 + 2λ2 m Zb , mϒn , mπ m2 ϒn , (30) where a4 + b4 + c4 − 2 a2b2 + a2c2 + b2c2 λ(a, b, c) = 2a The key component in Eq. (30) is the strong coupling gZbϒnπ . Relevant sum rules contain spectroscopic parameters of the tetraquark Zb and the mesons ϒ (n S) and π . The mass and current coupling of the Zb resonance have been calculated in the previous section. For numerical computations we take masses mϒn and decay constants fϒn of the mesons ϒ (n S) from Ref. [ 46 ]. The relevant information is shown in Table 1. In our calculations the Borel parameter M 2 and continuum threshold s0 are varied within the regions . M 2 = 10 − 13 GeV2, s0 = 124 − 128 GeV2, (31) which are almost identical to similar working windows in the mass and current coupling calculations being slightly shifted towards larger values. Parameters (32) (33) For the couplings gZbϒnπ we obtain (in GeV−1): gZbϒ1π = 0.019 ± 0.005, gZbϒ2π = 0.090 ± 0.031, For the widths of the decays Zb → ϒ (n S)π these couplings lead to the predictions The predictions obtained for the widths of the decays (Zb → ϒ (n S)π ) are the final results of this section and will be used for comparison with the experimental data. 4 Zb → hb(1 P )π and Zb → hb(2 P )π decays The second class of decays which we consider contains two processes Zb → hb(m P)π, m = 1, 2. We follow the same prescriptions as in the case of Zb → ϒ (n P)π decays and derive sum rules for the strong couplings gZbhbπ and gZbhbπ (hereafter we employ the short-hand notations hb ≡ hb(1 P) and hb ≡ hb(2 P)). From the analysis performed in the previous section it is clear that the corresponding sum rules will depend on numerous input parameters including mass and decay constant of the mesons hb(1 P) and hb(2 P). Information on the spectroscopic parameters of hb(1 P) is available in the literature. Indeed, in the context of the QCD sum rule method mass and decay constant of h(1 P) were calculated in Ref. [ 47 ]. But the decay constant fhb of the meson hb(2 P) was not evaluated; therefore, in the present work, we have first to find the parameters mhb and fhb , and we shall turn after that to our main task. 4.1 Spectroscopic parameters of the mesons hb(1 P) and hb(2 P) The meson h(1 P) is the spin-singlet P-wave bottomonium with quantum numbers J PC = 1+−, whereas h(2 P) is its first radial excitation. The parameters of the hb(1 P) and hb(2 P) mesons in the framework of QCD two-point sum rule method can be extracted from the correlation function μναβ ( p) = i d4x ei px 0 T Jμhν (x ) Jαhβ†(0) 0 , (34) where the interpolating current for hb(m P) mesons is chosen as Jμhν (x ) = bi (x )σμν γ5bi (x ). It couples both to hb(1 P) and hb(2 P), and is convenient for the analysis of J PC = 1+− mesons (see Ref. [ 47 ]). In order to find the required sum rules we use the “groundstate+first radial excitation+continuum” scheme. Then the physical side of the sum rule, Phys μναβ ( p) = 0 Jμhν |hb( p) hb( p)| Jαhβ†(0) 0 m2hb − p2 0| Jμhν hb( p) hb( p)| Jαhβ†(0) 0 contains two terms of interest and also a contribution of higher resonances and continuum states, denoted by dots. We continue by introducing the matrix elements 0| Jμν |h(b )( p) = fh( ) (εμ() pν − εν( ) pμ), h b and we recast the correlation function form Phys hb μναβ ( p) = m2hb − p2 f 2 gμα pν pβ − gμβ pν pα Phys μναβ ( p) into the (35) (36) (37) + m2 hb − p2 −gνα pμ pβ + gνβ pμ pα f 2 hb gμα pν pβ −gμβ pν pα − gνα pμ pβ + gνβ pμ pα , (38) where gμ(α) = −gμα + pμ pα m2h( ) b . The Borel transformation of Pμhνyαsβ ( p) can be obtained by simple replacements in Eq. (38): fh2( ) B m2h( ) − p2 = fh2(b ) e b b −m2h( ) /M2 b . The expression obtained in this way contains numerous Lorentz structures which, in general, may be employed to derive the sum rules for masses and decay constants: We choose a structure ∼ gμα pν pβ to extract sum rules. The term with the same structure should be isolated in the Borel transformation of μQνCαDβ ( p), i.e. in the expression of the correlation function calculated using the quark–gluon degrees of freedom. After simple computations for QCD μναβ ( p) we get QCD μναβ ( p) = i d4x ei px Tr γ5σαβ Sbji (−x ) ×σμν γ5 Sbij (x ) . The following operations are standard manipulations; they imply the Borel transforming of μQνCαDβ ( p), equating the structures ∼ gμα pν pβ in both the physical and the QCD sides of the equality obtained, and subtracting the continuum contribution. We obtain the second sum rule by acting on the first one by d/d(−1/M 2). These two sum rules allow us to evaluate the masses and decay constants of the hb(1 P) and hb(2 P) mesons. At the first stage we employ the “groundstate +continuum” scheme, which is commonly used in sum rule computations. This means that we include the excited hb(2 P) meson into the “higher resonances and continuum” part of sum rules and fix working windows for M 2 and s0. From these sum rules we extract spectroscopic parameters of the hb(1 P) meson mhb and fhb . At the next step we employ the same sum rules with s0∗ > s0 to include the contribution arising from hb(2 P), and we treat mhb and fhb evaluated at the first stage as fixed parameters. Numerical analysis restricts the variation of the parameters M 2 and s0 within the regions M 2 = 10 − 12 GeV2, s0 = 103 − 105 GeV2, and we find mhb = 9886+−8718 MeV, fhb = 325+−6517 MeV. At the next step we use The parameters of the hb(2 P) meson are among essentially new results of the present work; therefore, in Figs. 3 and 4 we demonstrate mhb(2P) and fhb(2P) as functions of the Borel parameter M 2 and the continuum threshold s0. 10.6 M 2 10 GeV2 M 2 11 GeV2 M 2 12 GeV2 M 2 10 GeV2 M 2 11 GeV2 M 2 12 GeV2 we see a reasonable agreement between them. 4.2 Widths of decays Zb → hb(1 P)π and Zb → hb(2 P)π Analysis of the vertices Zbhb(m P)π does not differ from the analogous investigation carried out in the previous section. We start here from the correlator μνλ( p, q) = i d4x ei px π(q)|T { Jμhν (x ) JλZb†(0)}|0 , and for its phenomenological representation get (42) . . . . (43) (44) Pμhνyλs( p, q) = 0 Jμhν hb ( p) p2 − m2hb × Zb( p )| JλZb†|0 Pμhνyλs( p, q) contains two terms of interest and contributions coming from higher resonances and continuum shown above as dots. Using matrix elements of the currents Jμhν and JλZb and introducing the vertex h(b ) ( p) π(q)|Zb( p ) = gZbh(b )π αβγ δεα∗ ( p)εβ ( p ) pγ pδ, we find The same correlation function expressed in terms of quark propagators takes the following form: μQνCλD( p, q) = Expanding uaαdβd in accordance with Eq. (24) and substituting into Eq. (46) the local matrix elements of the pion we obtain μQνCλD( p, q), which can be matched to Pμhνyλs( p, q) to fix the same tensor structures. In order to derive the sum rule we use the structures ∼ μλγ δ pγ pδ pν from both sides of the equality. The pion matrix element that contributes to this structure is 0|d(0)γ5γμu(0)|π(q) = i fπ qμ. In fact, it can be included into the chosen structure after replacement qμ = pμ − pμ. In the equality obtained we apply the soft limit q → 0 ( p = p ) and perform the Borel transformation on the variable p2. This operation leads to a sum rule for the two strong couplings gZbhbπ and gZbhbπ . The second expression is obtained from the first one by applying the operator d/d(−1/M 2). The principal output of these calculations, i.e. the spectral density ρhQCD(s), reads fπ ρQCD(s) = 12√2 h ρpert.(s) + ρn.−pert.(s) , where its perturbative part is given by the formula . The nonperturbative component of ρhQCD(s) includes contributions up to eight dimensions and has the form ρn.−pert.(s) = + gs3G3 m2b 0 Here the functions fk (z, s) are . In the expressions above the Dirac delta function δ(n)(s − is defined in accordance with ) δ(n)(s − dn ) = dsn δ(s − ). The widths of the decays Zb → hb(1 P)π and Zb → hb(2 P)π are calculated using the formula 2 (Zb → hb(m P)π ) = gZbhb(m P)π λ m Zb , mh(m P), mπ 12π In our numerical computations we employ the parameters of the hb(m P) mesons obtained in the previous subsection. The working regions of the Borel parameter M 2 and continuum threshold s0 are the same as in the analysis of the Zb → ϒ (n S)π decays. Below we provide our results for the strong couplings (in units GeV−1) gZbhbπ = 0.94 ± 0.27, gZbhbπ = 3.43 ± 0.93. (51) In Fig. 5 we plot the coupling gZbhbπ as a function of the Borel parameter and continuum threshold to show its dependence on these auxiliary parameters. It is easy to see that theoretical errors are within the limits accepted in sum rule calculations. Using Eq. (51) it is not difficult we evaluate the widths of the decays: (Zb → hb(1 P)π ) = 6.30 ± 1.76 MeV, (Zb → hb(2 P)π ) = 7.35 ± 2.13 MeV. (50) 3 (52) M 2 10.0 GeV2 M 2 11.5 GeV2 M 2 13.0 GeV2 0 10.0 The experimental data on the decay channels of the Zb(10610) resonance were studied and presented in a rather detailed form in Refs. [ 9–11 ]. Its full width was estimated as = 18.4 ± 2.4 MeV, an essential part of which, i.e. approximately 86% of , is due to the decay Zb → B+ B∗0+ B∗+ B0. The remaining part of the full width is formed by five decay channels investigated in the present work. It is clear that our results for the widths of the decays Zb → ϒ (n S)π and Zb → hb(n P)π overshoot the experimental data. Therefore, in the light of the present studies we refrain from an interpretation of the Zb(10610) resonance as a pure diquark– antidiquark [bu][bd] state. Nevertheless, theoretical predictions are encouraging for the ratios R(n) = (Zb → ϒ (n S)π ) (Zb → ϒ (1S)π ) , R(m) = (Zb → hb(m S)π ) , (Zb → ϒ (1S)π ) (53) where we normalize the widths of the different decay channels to (Zb → ϒ (1S)π ). The ratio R can be extracted from the available experimental data and can be calculated from the decay widths obtained in the present work. In order to fix existing similarities and differences between theoretical and experimental information on R we provide two sets of corresponding values in Table 2. It is worth to note that we use the latest available experimental information from Ref. [ 46 ]. Table 2 Experimental values and theoretical predictions for R R n = 2 n = 3 m = 1 m = 2 Exp. [ 46 ] 6.67+−32..1317 3.89−+12..5052 6.48−+23..1485 8.70−+34..4319 This work 12.63 ± 5.43 6.08 ± 2.76 4.63 ± 1.95 5.40 ± 2.32 It is seen that theoretical predictions follow the pattern of the experimental data: we observe the same hierarchy of theoretical and experimental decay widths. At the time, numerical differences between them are noticeable. Nevertheless, as a result of the large errors in both sets, there are sizable overlap regions for each pair of Rs, which demonstrates not only qualitative agreement between them but also the quantitative compatibility of the two sets. These observations may help one to understand the nature of the Zb resonance. The Belle Collaboration discovered two Zb and Zb resonances with very close masses. We have calculated the parameters of an axial-vector diquark–antidiquark state [bu][bd], and we interpreted it as Zb. It is possible to model the second Zb resonance using an alternative interpolating current, as has been emphasized in Sect. 2, and we explore its properties. The current with the same quantum numbers but different color organization may also play a role of such alternative (see, for example, Ref. [ 44 ]). One of the possible scenarios implies that the observed resonances are admixtures of these tetraquarks, which may fit measured decay widths. The diquark–antidiquark interpolating current used in the present work can be rewritten as a sum of molecular-type terms. In other words, some of the molecular-type currents effectively contribute to our predictions, and by enhancing these components (i.e. by adding them to the interpolating current with some coefficients) better agreement with the experimental data may be achieved. In other words, the resonances Zb and Zb may “contain” both the diquark– antidiquark and the molecular components. Finally, the Zb and Zb states may have pure molecular structures. But pure molecular-type bound states of mesons are usually broader than diquark–antidiquarks with the same quantum numbers and quark contents. In any case, all these suggestions require additional and detailed investigations. In the present study we have fulfilled only a part of this program. In the framework of QCD sum rule methods we have calculated the spectroscopic parameters of the Zb state by modeling it as a diquark–antidiquark state, and we found the widths of five of its observed decay channels. We have also evaluated the mass and decay constant of te hb(2 P) meson, which are necessary for analysis of the Zb → hb(2 P)π decay. Calculation of the Zb resonance’s dominant decay channel may be performed, for example, using the QCD three-point sum rule approach, which is beyond the scope of the present work. 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S. S. Agaev, K. Azizi, H. Sundu. Spectroscopic parameters and decays of the resonance \(Z_b(10610)\), The European Physical Journal C, 2017, 836, DOI: 10.1140/epjc/s10052-017-5421-5