Study of B B ¯ ∗ and B ∗ B ¯ ∗ interactions in I=1 and relationship to the Z b (10610) , Z b (10650) states
SStudy of B ¯ B ∗ and B ∗ ¯ B ∗ interactions in I = 1 and relationship tothe Z b (10610) , Z b (10650) states. J. M. Dias , , F. Aceti , E. Oset , Departamento de F´ısica Te´orica,Universidad de Valencia and IFIC,Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigaci´on de Paterna,Aptdo. 22085, 46071 Valencia, Spain Instituto de F´ısica,Universidade de S˜ao Paulo,C.P. 66318, 05389-970 S˜ao Paulo, SP, Brazil (Dated: September 11, 2018)
Abstract
We use the local hidden gauge approach in order to study the B ¯ B ∗ and B ∗ ¯ B ∗ interactions forisospin I=1. We show that both interactions via one light meson exchange are not allowed byOZI rule and, for that reason, we calculate the contributions due to the exchange of two pions,interacting and noninteracting among themselves, and also due to the heavy vector mesons. Then,to compare all these contributions, we use the potential related to the heavy vector exchange asan effective potential corrected by a factor which takes into account the contribution of the otherslight mesons exchange. In order to look for poles, this effective potential is used as the kernel ofthe Bethe-Salpeter equation. As a result, for the B ¯ B ∗ interaction we find a loosely bound statewith mass in the range 10587 − Z b (10610)reported by Belle Collaboration. For the B ∗ ¯ B ∗ case, we find a cusp at 10650 MeV for all spin J = 0 , , PACS numbers: 11.80.Gw, 12.38.Gc, 12.39.Fe, 13.75.Lb a r X i v : . [ h e p - ph ] O c t . INTRODUCTION In 2003 the Belle collaboration observed the first new charmoniumlike state called X (3872) in the B + → X (3872) K + → J/ψ π + π − K + process [1]. It was later confirmedby BaBar, CDF and D J/ψ plus pions which is anunusual property for a simple c ¯ c state. Furthermore, the predictions from potential modelsfor the mass and decay channels do not fit with the experimental results. For all thesereasons, a strong experimental and theoretical effort has been made in order to understandthe quark configuration of these new states as well as their production mechanisms, decaywidths, masses and spin-parity assignments. In Refs. [3–8] one can find a detailed discussionabout the current status of those states, commonly called X , Y and Z .Since the discovery of the X , Y and Z states, an enormous bulk of work has been donein an attempt to accomodate them in an exotic picture. By exotic we mean a more complexquark structure beyond quark-antiquark state, like hybrid, tetraquark, hadrocharmoniumand meson molecule. The exotic state idea is not new, actually is quite old, but beforethe discovery of Z + c (3900) by BESIII and Belle collaborations last year, no exotic structurehad been conclusively identified. It is a challenge to understand these new charmoniumlikestates as exotic since using the models mentioned above it is relatively simple to reproducethe masses of those states. The same challenges also concern the bottomoniumlike states.Among them, the Z b (10650) and Z b (10610) are very interesting. They were observed by theBelle collaboration in π ± h b ( nP ) and π ± Υ( mS ), with the n = 1 , m = 1 , ,
3, invariantmass distribution of the Υ(5 S ) decay channel [10]. As a result of the measurements, Bellereported: M Z b (10610) = (10608 . ± .
0) MeV, Γ Z b (10610) = (15 . ± .
5) MeV and for Z b (10650), M Z b (10650) = (10653 . ± .
5) MeV and Γ Z b (10650) = (14 . ± .
2) MeV. The quantum numbersare reported as J P = 1 + and positive G parity. The neutral partner has also been observedin the Υ(5 S ) → Υ( nS ) ππ decay in the belle Colaboration [11].In an attempt to understand the Z b (10610) and Z b (10650) configuration, some interpre-tations were considered. The authors of [12] treated the states as molecular states of B ¯ B ∗ and B ∗ ¯ B ∗ using HQSS, but the strength of the interaction was unknown. The proximityof the masses of these states to the B ¯ B ∗ and B ∗ ¯ B ∗ thresholds prompted the author of [13]to suggest that these peaks could be a consequence of cusps originated at these thresholds.This idea has been made more quantitative in a recent paper [14]. In [15] the dynamics ofhadro-quarkonium system was formulated, based on the channel coupling of a light hadron(h) and heavy quarkonium ( Q ¯ Q ) to intermediate open-flavor heavy-light mesons (Qq, Qq).In [16] the authors used QCD sum rules assuming tetraquarks or molecules, and in all casesthey could obtain good results, but the errors in the masses were of the order of 200 to 300MeV. In the same line, in [17] the states are also assumed to be tetraquarks. A tetraquarkpicture was assumed by the authors of Ref. [9], where using the framework of QCD sumrules, they calculated the Z b ’s mass, but the masses obtained were lower than those of the Z b states. In [18] the authors consider the states as molecular states driven by the onepion exchange interaction. In [19] heavy quark spin symmetry is used, analysing the powercounting of the loops, and concluding that the molecular nature of the states can account forthe observed features. In [20] the authors mention that using heavy quark spin symmetry2HQSS) and the molecular picture, states of 1 − should exist in addition to the reportedstates of 1 + . In [21] the molecular option is also supported by sum rules, but again withabout 220 MeV uncertainty in the mass. In [22] a tetraquark nature is invoked. In [23] B ¯ B ∗ , B ∗ ¯ B ∗ (in S-wave) are investigated in the framework of chiral quark models usingthe Gaussian expansion method. The bound states of B ¯ B ∗ , B ∗ ¯ B ∗ with quantum numbers I ( J P ) = 1(1 + ), which are good candidates for the Z b (10610) and Z b (10650) respectively,are obtained. Another B ¯ B ∗ bound state with I ( J P C ) = 0(1 ++ ), and other two B ∗ ¯ B ∗ with I ( J P C ) = 1(0 ++ ), I ( J P C ) = 0(2 ++ ) are predicted in that work. In [17] 1 + tetraquarks areinvoked and possible 1 ++ , 2 ++ states from charge conjugation are investigated. In [24] themolecular picture is again pursued and the Υ(5 S ) → Υ( nS ) π + π − decays are investigated.In [25] the authors make arguments of HQSS starting from the X(3872) extrapolating tothe beauty sector, and find a plausible molecular interpretation for the Z b (10610) state. In[26] once again the molecular structure is supported within HQSS. A different intepreta-tion is given in [27], where the initial pion emission mechanism is invoked to reproduce theΥ(5 S ) → Υ( nS ) ππ , with the second π and the resonance produced from the loop diagraminvolving three B ∗ states. Again from the molecular point of view in [28], several decaychannels are investigated in order to give support for the molecular picture. In [29], usingphenomenological Lagrangians and the hypothesis of molecular states, the Z → Υ( nS ) π transition rates are evaluated. Tetraquarks are again invoked in [30]. Pion exchange isconsidered in [31] and limits for the strength to produce binding are discussed. In [32] atetraquark is preferred, since meson exchange binds in I=0 but not in I=1. By using HQSSand assuming the states to be molecular states, different modes of production are evaluatedin [33].Using the chiral quark models, the authors of [34] interpret the states as loosely boundstates of B ¯ B ∗ , B ∗ ¯ B ∗ . Tetraquarks are again favoured in sum rules in [35]. In [36] theauthors use HQSS to relate these states, which are assumed to be molecular, to the X(3872).A molecular interpretation was again used in Ref. [37] in order to explain the states as B ∗ ¯ B and B ∗ ¯ B ∗ assuming a s- and d-wave mixture.The amount of theoretical work done is quite large, offering theoreticians a challengewith observed states that obviously cannot have a c ¯ c nature, which should have I = 0. Ourcontribution to the subject lines up with the molecular interpretation, using a dynamicalmodel that provides the strength of the interaction. We use for this purpose the extrapolationof the local hidden gauge approach to the heavy sector, extending results obtained for the Z c (3900) and Z c (4025) using that approach [38, 39] which at the same time was shown tofully respect the rules of HQSS [40, 41]. II. FORMALISM
In order to study B ¯ B ∗ and B ∗ ¯ B ∗ states, the extension of the local hidden gauge approach[42–44] to the heavy quark sector [45] seems most appropriate. The interaction is generatedby the exchange of a vector meson. If one exchanges light vectors the heavy quarks actas spectators and then, the heavy quark spin symmetry (HQSS) of QCD is automaticallyfulfilled [40]. However, following the approach of Refs. [38, 39], we can show that the B ¯ B ∗ and B ∗ ¯ B ∗ interactions by means one light meson exchange are not allowed by OZI rule for I = 1 states. In Fig. 1, a diagram illustrating an interaction between a B + ¯ B ∗ is shown. Inorder for this interaction to occur a d ¯ d state has to be converted into a u ¯ u state, which isOZI forbidden. This implies a cancellation between the contributions coming from ρ and ω π, η, η (cid:48) exchange in the limit of equalmasses for these mesons [38, 39]. B + ¯ B ∗ b ¯ b u ¯ d ¯ du FIG. 1. Diagram representing the B + ¯ B ∗ → B + ¯ B ∗ process through the exchange of q ¯ q , which isnot allowed by OZI rule. Because of this cancellation, we shall consider processes in which the OZI restriction nolonger holds. We, thus, calculate the contributions coming from heavy vector exchange andalso due to the exchange of two pions, interacting and non-interacting among themselves. A. B ¯ B ∗ and B ∗ ¯ B ∗ interactions via heavy vector exchange In order to evaluate B ¯ B ∗ and B ∗ ¯ B ∗ interactions due to the exchange of vector mesons,we need the Lagrangians describing the V P P and
V V V vertices, namely L V P P = − ig (cid:104) V µ [ P, ∂ µ P ] (cid:105) , (1) L V V V = ig (cid:104) ( V µ ∂ ν V µ − ∂ ν V µ V µ ) V ν (cid:105) . (2)The coupling g is given by g = M V / f π , being f π = 93 MeV the pion decay constant, while M V is the vector meson mass.In Eqs. (1) and (2), the symbol (cid:104) (cid:105) stands for the trace of SU(4). The vector field V µ isrepresented by the SU(4) matrix, which is parametrized by 16 vector mesons including the15-plet and singlet of SU(4), V µ = ω √ + ρ √ ρ + K ∗ + ¯ B ∗ ρ − ω √ − ρ √ K ∗ B ∗− K ∗− ¯ K ∗ φ B ∗− s B ∗ B ∗ + B ∗ + s J/ψ µ , (3)where the ideal mixing has been taken for ω , φ and J/ψ . On the other hand, P is a matrixcontaining the 15-plet of the pseudoscalar mesons written in the physical basis in which η ,4 (cid:48) mixing is taken into account [46], P = η √ + η (cid:48) √ + π √ π + K + ¯ B π − η √ + η (cid:48) √ − π √ K B − K − ¯ K − η √ + (cid:113) η (cid:48) B − s B B + B + s η b . (4)The channels we are interested in are those with B = 0, S = 0 and isospin I = 1. In the B ∗ ¯ B ∗ case, they are B ∗ ¯ B ∗ and ρ Υ. In the case of B ¯ B ∗ we are only interested in the positive G -parity combination, namely ( B ¯ B ∗ + cc ) / √ η b ρ and π Υ. B ∗ ¯ B ∗ case B ∗ + ( k , ǫ ) B ∗− ( k , ǫ ) B ∗− ( k , ǫ ) B ∗ + ( k , ǫ ) B ∗− ( k , ǫ ) B ∗ ( k , ǫ ) B ∗ +( k , ǫ ) B ∗ ( k , ǫ ) B ∗ ( k , ǫ ) B ∗ ( k , ǫ )¯ B ∗ ( k , ǫ ) ¯ B ∗ ( k , ǫ ) ρ , ω, Υ( k − k , ǫ (0) ) ρ + ( k − k , ǫ (0) ) ρ , ω, Υ( k − k , ǫ (0) ) FIG. 2. Vector exchange diagrams contributing to the process B ∗ ¯ B ∗ → B ∗ ¯ B ∗ . Consider now the reaction B ∗ ¯ B ∗ → B ∗ ¯ B ∗ . Here we are following the same steps as inRef. [45], in which the authors were concerned in the D ∗ ¯ D ∗ case. As in [45] we also considerthat the external vectors have negligible three-momentum with respect to their masses. Inour case, the most important diagrams are depicted in Fig. 2. As an example, we shallcalculate in detail the amplitude of the first diagram in Fig. 2. The evaluation of the otherones is analogous. For this end, we must calculate the three-vector vertex which is given bythe Lagrangian of Eq. (2). Figs. 3(a) and (b) illustrate the three-vector vertices B ∗ + ¯ B ∗ + ρ and B ∗− ¯ B ∗− ρ with the momenta assignments. The corresponding vertex functions are t B ∗ + B ∗ + ρ = g √ k + k ) µ (cid:15) ν (cid:15) ν (cid:15) (0) µ , (5) t B ∗− B ∗− ρ = g √ k + k ) µ (cid:15) ν (cid:15) ν (cid:15) (0) µ . (6)Once we have determined the vertices, it is possible to calculate the amplitude for thefirst diagram of Fig. 2. Considering all the particles involved in the exchange, we obtain t B ∗ + B ∗− → B ∗ + B ∗− = − g (cid:20) M + 1 M ρ + 1 M ω (cid:21) ( k + k ) · ( k + k ) (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν , (7)where M Υ , M ρ and M ω are the masses of the Υ, ρ and ω mesons, respectively.5 ∗ + ( k , ǫ ) B ∗ + ( k , ǫ ) ρ ( k − k , ǫ (0) ) B ∗− ( k , ǫ ) B ∗− ( k , ǫ ) ρ ( k − k , ǫ (0) )( a ) ( b ) FIG. 3. Three-vector vertex associated with B ∗ + B ∗ + ρ . As we are interested in the B ∗ ¯ B ∗ interaction in the I = 1 channel, we must rewrite Eq.(7) in the isospin basis. The isospin states are | B ∗ ¯ B ∗ (cid:105) I =1 = − √ | B ∗ + ¯ B ∗− (cid:105) + 1 √ | B ∗ ¯ B ∗ (cid:105) , (8) | B ∗ ¯ B ∗ (cid:105) I =0 = 1 √ | B ∗ + ¯ B ∗− (cid:105) − √ | B ∗ ¯ B ∗ (cid:105) . By taking into account all the three diagrams of Fig. 2, we get t I =1 B ∗ ¯ B ∗ → B ∗ ¯ B ∗ = g (cid:20) M ρ M ω + M ( − M ω + M ρ )2 M M ω M ω (cid:21) ( k + k ) · ( k + k ) (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν , (9)which shows explicitly the cancellation of ρ and ω exchange.In order to rewrite the amplitude given by Eq. (9) in terms of spin 0, 1 and 2 states, weuse the spin projectors P (0) , P (1) and P (2) given by [45] P (0) = 13 (cid:15) µ (cid:15) µ (cid:15) ν (cid:15) ν , P (1) = 12 ( (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν − (cid:15) µ (cid:15) ν (cid:15) ν (cid:15) µ ) , P (2) = 12 ( (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν + (cid:15) µ (cid:15) ν (cid:15) ν (cid:15) µ ) − (cid:15) µ (cid:15) µ (cid:15) ν (cid:15) µ , (10)where the order of the particles 1, 2, 3 and 4 is implicit. In terms of those projectors thepolarization vector combination (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν appearing in Eq. (9) is equal to (cid:15) µ (cid:15) ν (cid:15) µ (cid:15) ν = P (0) + P (1) + P (2) . (11)Therefore, substituting Eq. (11) into Eq. (9), projecting it in s-wave, and including thecontact term already evaluated in Ref. [45], we obtain t I =1 ,S =0 , , B ∗ ¯ B ∗ → B ∗ ¯ B ∗ = − g + g (cid:20) M ρ M ω + M ( − M ω + M ρ )4 M M ω M ρ (cid:21) (4 M B ∗ − s ) , (12)where s stands for the center of mass energy of the B ∗ ¯ B ∗ system.6onsider now the other channel, B ∗ ¯ B ∗ → ρ Υ. The most relevant diagrams are depictedin Fig 4. The procedure to get the amplitude for this channel is analogous to what we havedone earlier. Thus, the amplitude in isospin I = 1 basis for the spin S = 0 , t I =1 ,S =0 , B ∗ ¯ B ∗ → ρ Υ = − g + g (cid:20) M B ∗ + M + M ρ − sM B ∗ (cid:21) . (13)The interaction in S = 1 vanishes as a consequence of a cancellation of terms where the ρ and Υ are interchanged in the diagrams. The diagonal ρ Υ → ρ Υ transition is again OZIforbidden and null in this approach. B ∗ + ( k , ǫ ) ρ ( k , ǫ ) B ∗− ( k , ǫ ) Υ( k , ǫ ) B ∗ + ( k − k , ǫ (0) ) ρ ( k , ǫ ) B ∗ ( k , ǫ )¯ B ∗ ( k , ǫ ) Υ( k , ǫ )¯ B ∗ ( k − k , ǫ (0) ) FIG. 4. Vector exchange diagrams contributing for the B ∗ ¯ B ∗ → ρ Υ channel.
Eqs. (12) and (13) will be used as a kernel of the Bethe-Salpeter equation as we shalldiscuss it later. B ¯ B ∗ case In this case, the Lagrangians defined in Eqs. (1) and (2) can also be used to provide thevertices of the
P V → P V interaction through exchange of a heavy vector. The resultingamplitudes were already calculated in s-wave in Refs. [47, 48]. In particular the authorswere concerned with axial-vector resonances dynamically generated. Yet, in Ref. [38] thesame equation for the amplitude is used in order to study D ¯ D ∗ interaction. Here, we extendthese amplitudes for the B ¯ B ∗ interaction in the isospin I = 1 channel, with the result V ij ( s ) = − (cid:126)(cid:15) (cid:126)(cid:15) (cid:48) f π C ij (cid:20) s − ( M + m + M (cid:48) + m (cid:48) ) − s ( M − m )( M (cid:48) − m (cid:48) ) (cid:21) , (14)where the masses M ( M (cid:48) ) and m ( m (cid:48) ) in Eq. (14) correspond to the initial (final) vectormeson and pseudoscalar meson, respectively. The indices i and j represent the initial andfinal V P channels ( B ¯ B ∗ + cc ) / √ η b ρ and π Υ.The C ij are elements of a 3 × G -parity of the B ¯ B ∗ combination, is defined as C ij = − ψ √ γ √ γ √ γ √ γ , (15)where γ = (cid:16) m L m H (cid:17) and ψ = (cid:16) m L m H (cid:48) (cid:17) . Those factors are defined in this way in order to takeinto account the suppression due to the exchange of a heavy vector meson. Concerning the7arameters m L , m H and m H (cid:48) , we choose their values in order to have the same order ofmagnitude of the light and heavy vector meson masses: m L = 800, m H = 5000 MeV and m H (cid:48) = 9000 MeV.
3. The T -Matrix The results of the amplitudes discussed earlier provide the potential or kernel to be usedin the Bethe-Salpeter equation in coupled channels, T = (1 − V G ) − V , (16)where V is the potential, which in the B ∗ ¯ B ∗ case is a 2 × B ∗ ¯ B ∗ and ρ Υ. In the case of B ¯ B ∗ , V is a 3 × C ij defined by Eq. (15), associated with the channels B ¯ B ∗ , η b ρ and π Υ.In Eq. (16), G is a diagonal matrix and its elements are given by the two meson loopfunction, G l for each channel l : G l = i (cid:90) d q (2 π ) q − m + i(cid:15) q − P ) − M + i(cid:15) , (17)where m is the mass of the pseudoscalar (in the B ¯ B ∗ case) or vector (in the B ∗ ¯ B ∗ case),while M is the vector meson mass involved in the loop in the channel l . In Eq. (17) P means the total four-momentum of the mesons. The integral of Eq. (17) is logarithmicallydivergent and it can be regularized with a cut off in the momentum space or dimensionalregularization. With the cut off method G l = q max (cid:90) d q (2 π ) ω + ω ω ω P ) − ( ω + ω ) + i(cid:15) , (18)where ω = (cid:112) m + (cid:126)q and ω = (cid:112) M + (cid:126)q and q max is a free parameter. In dimensionalregularization there is a scale µ and a subtraction constant α ( µ ) acting as a free parameter,namely, G l = 116 π ( α l + log m µ + M − m + s s log M m + p √ s (log s − M + m + 2 p √ s − s + M − m + 2 p √ s + log s + M − m + 2 p √ s − s − M + m + 2 p √ s )) . (19)with p standing for the three-momentum of the mesons in the center-of-mass frame.For the sake of comparison of the different potentials obtained, it is interesting to recallthat Eq. (16) with the cut off regularization of Eq. (18) can be obtained from the Lippmann-Schwinger equation using a potential in momentum space [49] V ( (cid:126)q, (cid:126)q (cid:48) ) = V θ ( q max − | (cid:126)q | ) θ ( q max − | (cid:126)q (cid:48) | ) . (20)Hence, assuming (cid:126)q ≈ (cid:126)q (cid:48) can play the role of momentum transferin loop diagrams, and then V as a function of (cid:126)q (cid:48) remains constant up to q max , where it goesto zero. 8 . The σ exchange contribution to the B ¯ B ∗ and B ∗ ¯ B ∗ interactions The potential due to the σ exchange in some cases provides an important contributionto the interaction. In Ref. [50] the authors studied the N N system considering that the σ resonance arises from the interaction of two pions, providing an important contributionto the binding energy for the N N system. In Refs. [38, 39] the same idea was applied tothe D ¯ D ∗ and D ∗ ¯ D ∗ cases. Following the approach of those references we shall extend theformalism to the bottom sector, more specifically, to study the B ¯ B ∗ and B ∗ ¯ B ∗ interactions. B ∗ + B ∗ + B + π π π π ¯ B ∗ ¯ B ∗ ¯ B a ) B ∗ + B ∗ + B + π π π − π − ¯ B ∗ ¯ B ∗ B − b ) B ∗ + B ∗ + B π − π − π π ¯ B ∗ ¯ B ∗ ¯ B c ) B ∗ + B ∗ + B π − π − π − π − ¯ B ∗ ¯ B ∗ ¯ B − d ) FIG. 5. Diagrams contributing to the two pions interaction in lowest order in I = 1 for the B ∗ ¯ B ∗ → B ∗ ¯ B ∗ process. Let us consider first, the B ∗ ¯ B ∗ case. The diagrams contributing to this interaction areillustrated in Fig. 5. As can be seen from Fig. 5, each diagram has four vertices containingtwo pseudoscalars, the π and B ( ¯ B ) mesons and one B ∗ ( ¯ B ∗ ) vector. Their evaluation isdone by means of the local hidden gauge Lagrangian already defined in Eq. (1). On theother hand, instead of calculating the vertices and then the amplitude from the Lagrangianof Eq. (1), we start from the amplitude obtained in Ref. [39] and substitute the masses ofthe D and D ∗ mesons by the masses of the B and B ∗ , respectively. As a result, we obtain − it σB ∗ ¯ B ∗ = − i V t I =0 ππ → ππ , (21)where t I =0 π π → π π is the isoscalar amplitude for the π π interaction, namely t I =0 π π → π π = − f s (cid:48) − m π f G ( s (cid:48) )( s (cid:48) − m π ) , (22)with G ( s (cid:48) ) the two pions loop function suited to this case (whose explicit form is given in[39, 50]) and with P the total π π momentum, with the pions travelling to the right in thediagrams. Hence, P = s (cid:48) is actually the variable t for the B ∗ + ¯ B ∗ system.In Eq. (21), V is a factor that takes into account the contributions coming from thetriangular loops of the diagram. The detailed derivation of the V factor can be found in9ef. [39]. We adopt the Breit frame, p ≡ ( p , (cid:126)q/ ,p (cid:48) ≡ ( p (cid:48) , − (cid:126)q/ ,p ≡ ( p , (cid:126)p ) , (23)where (cid:126)q is the three-momentum transferred in the process and p and p (cid:48) the momenta forthe two incoming B ∗ . The equations for V are obtained from [39] with the trivial changesin the masses of the particles. It also contains the factor ( M B ∗ /M K ∗ ) replacing the factor( M D ∗ /M K ∗ ) in [39] as demanded by HQSS in [40]. q H MeV L t Σ FIG. 6. Potential t σB ∗ ¯ B ∗ as a function of the momentum transferred in the process. Finally, substituting Eq. (22) into Eq. (21) and taking s = − (cid:126)q since there is no energyexchange, we get the following expression for the potential t σB ∗ ¯ B ∗ ( (cid:126)q ) = V
32 1 f (cid:126)q + m π − G ( − (cid:126)q ) f ( (cid:126)q + m π ) , (24)with V = (cid:15) µ (cid:15) (cid:48) ν ( ag µν + cp (cid:48) µ p ν ) (25)and a and c also given in [39] with trivial changes in the masses. Assuming the spatialcomponents of the momenta p µ and p (cid:48) ν smaller than the vector masses, which impliestaking (cid:15) = 0, only the term with the a coefficient contributes to the potential, providingthe (cid:15) (cid:15) (cid:48) combination. The other vertex gives the same structure and then we have the (cid:15) (cid:15) (cid:48) (cid:15) (cid:15) (cid:48) combination. Hence, the potential can be rewritten as t σB ∗ ¯ B ∗ ( (cid:126)q ) = a (cid:34) f (cid:126)q + m π − G ( − (cid:126)q ) f ( (cid:126)q + m π ) (cid:35) (cid:15) µ (cid:15) (cid:48) ν (cid:15) µ (cid:15) (cid:48) ν , (26)where we have rewritten the polarization vectors combinations in order to associate thesubindices 1 , , → t σB ∗ ¯ B ∗ ( (cid:126)q ) = a (cid:34) f (cid:126)q + m π − G ( − (cid:126)q ) f ( (cid:126)q + m π ) (cid:35) ( P (0) + P (1) + P (2) ) . (27)In Fig. 6 we can see the plot of the t σB ∗ ¯ B ∗ potential, Eq. (27), as a function of the transferredmomentum (cid:126)q . B + B + B ∗ + π π π π ¯ B ∗ ¯ B ∗ ¯ B a ) B + B + B ∗ + π π π − π − ¯ B ∗ B − b ) B + B + B ∗ π − π − π π ¯ B ∗ ¯ B ∗ ¯ B c ) B + B + B ∗ π − π − π − π − ¯ B ∗ ¯ B ∗ ¯ B − d )¯ B ∗ FIG. 7. Diagrams contributing to the two pion exchange interaction in lowest order for the B ¯ B ∗ → B ¯ B ∗ process in I = 1. Next, we shall consider the same mechanism, but now for the B ¯ B ∗ case. The diagramsfor this process are shown in Fig. 7. For this case, the potential t σB ¯ B ∗ has a difference incomparison with the former case. Now we have two different triangular loops. This impliestwo V factors in Eq. (21), where each factor is associated with each triangular loop. Hence,the potential t σB ¯ B ∗ is given by − it σB ¯ B ∗ = − i V ¯ V t I =0 ππ → ππ , (28)where t I =0 ππ → ππ is the isoscalar amplitude defined in Eq. (22) and ¯ V is again given by Eq.(29)in [39] with trivial changes of masses. The potential t σB ¯ B ∗ is plotted in Fig. 8. C. The exchange due to the two uncorrelated pions
In this case, the pions are not interacting, then only the diagrams a) and d) of Figs. 5and 7 contribute for the B ∗ ¯ B ∗ and B ¯ B ∗ interactions. Details on the evaluation can be foundin [39]. The amplitude t π πB ∗ ¯ B ∗ can be rewritten in terms of its spin components as t π πB ∗ ¯ B ∗ = 54 g B A (cid:90) d p (2 π ) (4 (cid:126)p − (cid:126)q F ω + ω ω ω E B p − ω − E B + i(cid:15) × p − ω − E B + i(cid:15) (cid:18) E B + ω + ω − p p − ω − E B + i(cid:15) + E B + ω + ω − p p − ω − E B + i(cid:15) (cid:19) , (29)11
500 1000 1500 - - - - - q H MeV L t Σ BB * FIG. 8. Potential t σB ¯ B ∗ as a function of the momentum transferred in the process. p p ′ p p − p ′ p − p p ′ p p − p ′ + p (a) p p ′ p p − p ′ p − p p ′ p p − p ′ + p (b) FIG. 9. Momenta assignments in the two uncorrelated pion exchange in B ∗ ¯ B ∗ → B ∗ ¯ B ∗ and B ¯ B ∗ → B ¯ B ∗ , respectively. where A = 5 is associated with spin J = 0, while A = 2 is related to the J = 2 case, ω = (cid:112) ( (cid:126)p + (cid:126)q/ + m π , ω = (cid:112) ( (cid:126)p − (cid:126)q/ + m π are the energies of the pions and E B ( (cid:126)p ) = (cid:112) (cid:126)p + m B is the energy of the B meson. F ( (cid:126)q ) is a form factor of the type F = F ( (cid:126)p + (cid:126)q F ( (cid:126)p − (cid:126)q Λ + ( (cid:126)p + (cid:126)q ) Λ Λ + ( (cid:126)p − (cid:126)q ) , (30)with Λ = 700 GeV, which is also used later to help the convergence. Note that, accordingto [40], the coupling g = M V / f π used in Sec. II A is now replaced by g B = ( M B ∗ /M K ∗ ) g to account for the requirements of heavy quark spin symmetry. On the other hand, thiscorrection is automatically implemented in the extrapolation of the vector exchange to theheavy sector (Weinberg-Tomozawa term) because this term is explicitly proportional to theexternal B ∗ energies. 12
500 1000 1500 2000 - - - - - q H MeV L t box B * B * FIG. 10. Potential t B ∗ ¯ B ∗ ππ for non-interacting pion exchange in the case of J = 0 (solid line) and J = 2 (dashed line). In Fig. 10 we can see the amplitude for the two spin cases as a function of the momentumtransfer.For the B ¯ B ∗ case we find t π πB ¯ B ∗ = − g B (cid:126)(cid:15) (cid:48) (cid:126)(cid:15) (cid:48)(cid:48) (cid:90) d p (2 π ) ( (cid:126)p − (cid:126)q ) (cid:20) (4 (cid:126)p − (cid:126)q − (cid:126)q (cid:20) (2 (cid:126)p (cid:126)q ) − (cid:126)q (cid:21)(cid:21) F ω + ω ω ω × E B E V [ ω + ω + ω ω − ( ω + ω )(2 p − E B ∗ − E B ) + ( p − E B ∗ )( p − E B )] × p − ω − E B ∗ + i(cid:15) p − ω − E B + i(cid:15) p − ω − E B ∗ + i(cid:15) p − ω − E B + i(cid:15) , (31)where E B ∗ ( (cid:126)p ) = (cid:112) (cid:126)p + m B ∗ is the energy of the B ∗ meson. The amplitude t π πB ¯ B ∗ as afunction of the momentum transfer is plotted in Fig. 11 (cid:45) (cid:45) (cid:45) q (cid:72) MeV (cid:76) t ΠΠ BB (cid:42) FIG. 11. Potential t π πB ¯ B ∗ for non-interacting pion exchange as a function of the momentum trans-ferred in the process. II. ITERATED EXCHANGE OF TWO LIGHT MESONS
In this section we evaluate the contribution coming from the iterated exchange of twolight mesons, shown in Fig. 12 in the case of the B ∗ ¯ B ∗ (a) and of the B ¯ B ∗ (b) interactions. π, η, η ′ π, η, η ′ B ∗ + B ∗ + ¯ B ∗ ¯ B ∗ B + ¯ B π, η, η ′ π, η, η ′ B ∗ + B ∗ + ¯ B ¯ B B + ¯ B ∗ ( a ) ( b ) FIG. 12. Iterated exchange of two light mesons for the B ∗ ¯ B ∗ (a) and B ¯ B ∗ (b) cases. In the case of B ∗ ¯ B ∗ , the details of the calculation can be found in Sec. C of Ref. [39]and they lead to the following expression for the amplitude: t boxB ∗ ¯ B ∗ = 14 t boxππ + 19 t boxηη + 136 t boxη (cid:48) η (cid:48) − t boxπη − t boxπη (cid:48) + 19 t boxηη (cid:48) , (32)where t boxij = g B S J (cid:90) d p (2 π ) (cid:126)p F m D ∗ + ω − E B ( (cid:126)p ) ± i(cid:15) m B ∗ + ω − E B ( (cid:126)p ) ± i(cid:15) × E B ( (cid:126)p )) (cid:16) ω ω ω + ω Num m B ∗ − ω − E B ( (cid:126)p ) + i(cid:15) m B ∗ − ω − E B ( (cid:126)p ) + i(cid:15) + 1 E B ( (cid:126)p ) − m B ∗ + ω + i(cid:15) E B ( (cid:126)p ) − m B ∗ + ω + i(cid:15) m B ∗ − E B ( (cid:126)p ) + i(cid:15) (cid:17) , (33)where i j = π, η, eta (cid:48) and ω and ω are their energies of the two light mesons exchanged, S J = J = 0 J = 2 , (34)and Num = − ( ω + ω + ω ω ) + ( m B ∗ − E B ( (cid:126)p )) . (35)14he former calculation has been done at threshold. The momentum transfer dependence on (cid:126)q can be obtained easily from Eq. (33) by taking for the initial and final states four-momenta p = ( p , (cid:126)q/ p = ( p , − (cid:126)q/ p = ( p , − (cid:126)q/
2) and p = ( p , (cid:126)q/
2) ( p , p momenta of thetwo final B ∗ ).In Fig. 13 the amplitude t boxB ∗ ¯ B ∗ is plotted as a function of the momentum transferred (cid:126)q for the case J = 0 (dashed line) and J = 2 (solid line). q (cid:72) MeV (cid:76) t box B (cid:42) B (cid:42) FIG. 13. Amplitude t boxB ∗ ¯ B ∗ as a function of the momentum transferred in the process for the case J = 0 (dashed line) and J = 2 (solid line). With a similar procedure we can obtain the amplitude in the case of B ¯ B ∗ , using againthe Lagrangian of Eq. (1). We find t boxB ¯ B ∗ = 14 ˜ t boxππ + 19 ˜ t boxηη + 136 ˜ t boxη (cid:48) η (cid:48) −
13 ˜ t boxπη −
16 ˜ t boxπη (cid:48) + 19 ˜ t boxηη (cid:48) , (36)where ˜ t boxij = g B (cid:126)(cid:15) · (cid:126)(cid:15) (cid:48) (cid:90) d p (2 π ) (cid:126)p F E B ∗ ( (cid:126)p ) 1 E B ( (cid:126)p ) 1 E B ∗ ( (cid:126)p ) + ω − E B ( (cid:126)p ) ± i(cid:15) × E B ∗ ( (cid:126)p ) + ω − E B ( (cid:126)p ) ± i(cid:15) E B ∗ ( (cid:126)p ) + ω − E B ( (cid:126)p ) ± i(cid:15) (cid:16) ω ω × ω + ω E B ( (cid:126)p ) − ω − E B ∗ ( (cid:126)p ) + i(cid:15) Num (cid:48) E B ( (cid:126)p ) − ω − E B ∗ ( (cid:126)p ) + i(cid:15) + 1 E B ( (cid:126)p ) + ω − E B ∗ ( (cid:126)p ) − i(cid:15) E B ( (cid:126)p ) + ω − E B ∗ ( (cid:126)p ) − i(cid:15) × M B − E B ( (cid:126)p ) + M B ∗ − E B ∗ ( (cid:126)p ) + i(cid:15) (cid:17) , (37)with i, j = π, η, η (cid:48) . The numerator Num (cid:48) in Eq. (37) is given byNum (cid:48) = − ( ω + ω + ω ω ) + ( ω + ω )( M B + E B − M B ∗ − E B ∗ ) × ( M B − E B ∗ )( M B ∗ − E B ) . (38) E B , E B ∗ , ω and ω are already defined in Section II C.The potential t boxB ¯ B ∗ is plotted in Fig. 14 as a function of the tranferred momentum (cid:126)q .15
500 1000 1500 2000 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) q (cid:72) MeV (cid:76) t box BB (cid:42) FIG. 14. Amplitude t boxB ¯ B ∗ as a function of the momentum transferred in the process. IV. RESULTS - - - q H MeV L t FIG. 15. Comparison between the potentials t B ∗ ¯ B ∗ → B ∗ ¯ B ∗ (small dashed line, vector exchange Eq.(12)), t σB ∗ ¯ B ∗ (dotted line, Eq. (27)), t ππB ∗ ¯ B ∗ for J = 0 (dotted dashed line, Eq. (29)) and J = 2(solid line), t boxB ∗ ¯ B ∗ for J = 0 (solid thick line, Eq. (32)) and J = 2 (large dashed line) as functionsof the momentum transferred in the process. After we have calculated the amplitudes of all the processes contributing to the B ∗ ¯ B ∗ and B ¯ B ∗ interactions, we want to make a rough estimate of the strength of each potential. Thisis done by evaluating the integral (cid:82) d q V ( q ) in order to take into account the contributionscoming from the exchange of light mesons and use them to obtain an effective potential V eff . We will follow a simple strategy to account for the different potentials. We will getthe strength (cid:82) d q V i ( q ) for all the potentials exchanging light mesons and sum them. Thenwe convert the sum into an effective potential of the type of the vector exchange, V eff θ ( q max − | (cid:126)q | ) θ ( q max − | (cid:126)q (cid:48) | ) , (39)16
500 1000 1500 2000 - - - - - q H MeV L t FIG. 16. Comparison between the potentials t B ¯ B ∗ → B ¯ B ∗ (solid line, Eq. (14)), t σB ¯ B ∗ (dashed line,Eq. (28)), t ππB ¯ B ∗ (dotted line, Eq. (31)), t boxB ¯ B ∗ (dotted dashed line, Eq. (36)) as functions of themomentum transferred in the process. where q max is the maximum momentum used in the loops in Eq. (18) (see Eq. (20)), suchthat (cid:82) q 2. For the case of B ¯ B ∗ the factor varies between 30 and 64.In the following we study the shape of | T | for both B ∗ ¯ B ∗ and B ¯ B ∗ cases. As wewill discuss in detail, both amplitudes show a clear peak and the large uncertainties onthe potential do not affect drastically its position, which justifies a posteriori the approachfollowed indulging in large uncertainties. A. B ¯ B ∗ case In this case, we are interested in studying the T matrix for the channels: B ¯ B ∗ , η b ρ and π Υ. We evaluated the transition matrix T between those channels for values of √ s around10600 MeV. In order to do this, we use the dimensional regularization formula for the loopfunction G , given by Eq. (19). To obtain reasonable values of the subtraction constantsin each channel we proceed as follows: we take a cut off q max , then we find a subtractionconstant that provides at threshold the same G function obtained with the cut off method.Here we are taking q max = 700 MeV, for which we find α B ¯ B ∗ = − . α η b ρ = − . 56 and α π Υ = − . T matrix due to the variation17 000 9000 10 000 11 000 12 000050 000100 000150 000200 000 s (cid:64) MeV (cid:68) (cid:200) T FIG. 17. | T | as a function of the √ s center of mass energy for the case of B ¯ B ∗ . Each curve isassociated with a value of the integration limit: 700 MeV, 800 MeV, 900 MeV, 1000 MeV, 1100MeV. The peak moves from right to left as the integration limit increases. of q max were studied. However, in the current case, the changes due to this parameter aresmaller than the ones due to the variations of the upper limit of the integral (cid:82) d q V ( q )used to estimate V eff . In Fig. 17 the shape of | T | , the component of the T matrixthat describes the transition B ¯ B ∗ → B ¯ B ∗ , for different values of the integration limit, isdepicted. As can be seen, even choosing values of the limit between 700 and 1100 MeV,the effect on the binding and the width is small. As a result, we find that the position ofthe peak moves slightly to higher energies for decreasing values of the upper limit and it isseen in the range of 10587 − M Z b (10610) = (10608 . ± . 0) MeV. It is worth noting that both the η b ρ and π Υ channels are open for decays, and this gives a width between 1 . Z b (10610) = (15 . ± . 5) MeV. B. B ∗ ¯ B ∗ case For this case we have two channels: B ∗ ¯ B ∗ and ρ Υ. Again, we use the dimensionalregularization form of the loop function G , Eq. (19), with µ = 1500 MeV and the subtractionconstants α B ∗ ¯ B ∗ = − . 79 and α ρ Υ = − . 56, corresponding to a cut off value equal to q max = 700 MeV.Fig. 18 shows the shape of | T | , which means the component of the T matrix thatdescribes the transition from B ∗ ¯ B ∗ to itself, for different values of the integration limitplotted as a function of the center of mass energy, √ s , of the system. This peak correspondsto spin J = 0. In Fig 19, we show the shape of | T | for the J = 2 case, again for differentvalues of the integration limit. It is important to emphasize that, according to Eq. (13),there is no contribution in the transition matrix T from B ∗ ¯ B ∗ to ρ Υ channel for spin J = 1.In this case, B ∗ ¯ B ∗ stands as a single channel.18 s (cid:64) MeV (cid:68) (cid:200) T FIG. 18. | T | as a function of the √ s center of mass energy for the case of B ∗ ¯ B ∗ for J = 0. Eachcurve is associated with a value of the integration limit: 700 MeV, 800 MeV, 900 MeV, 1000 MeV,1100 MeV. The peak moves from bottom to top as the integration limit increases. 10 400 10 500 10 600 10 700 10 800 10 900 11 0000200040006000800010 000 s (cid:64) MeV (cid:68) (cid:200) T FIG. 19. | T | as a function of the √ s center of mass energy for the case of B ∗ ¯ B ∗ for J = 2. Eachcurve is associated with a value of the integration limit: 700 MeV, 800 MeV, 900 MeV, 1000 MeV,1100 MeV. The peak moves sligthtly from bottom to top as the integration limit increases. From these figures we can see that the variations of the integration limit cause no effectto the peak position, as we already noted in the B ¯ B ∗ case. It is interesting to note that,even with the large uncertainties in the potential admitted, we always find a structure forthe peak of | T | which corresponds clearly to a cusp. Whether to call this a resonant stateor not it is a question of criterion. We should however note that the a (980) appears in theexperiments (or in the theories) [53, 54] as a cusp and is universally accepted as a resonance.19ur findings, obtained a cusp for the | T | amplitude in this case, would come to supportthe claims of the former works [13, 14].For the sake of completeness, we repeat the calculation considering the spin J = 1 case.Here we have a single channel problem, T = ˜ t B ∗ ¯ B ∗ → B ∗ ¯ B ∗ − ˜ t B ∗ ¯ B ∗ → B ∗ ¯ B ∗ G B ∗ ¯ B ∗ , (40)where G B ∗ ¯ B ∗ is the loop function defined by Eq. (19) for the B ∗ ¯ B ∗ channel, while ˜ t B ∗ ¯ B ∗ → B ∗ ¯ B ∗ is the B ∗ ¯ B ∗ → B ∗ ¯ B ∗ vector exchange potential already defined in Eq. (12), plus thecontribution from V eff due to the exchange of two interacting pion exchange. In this case, wesaw that the noninteracting pion exchange vanished, and the interacting two pion exchangewas also small (see Fig. 6), smaller than the vector exchange (see Fig. 15), in all range.This is why, in this case, in order to play with uncertainties we follow the strategy of Refs.[38, 39] and we change the range of the vector exchange potential, by changing the cut off q max to values from 700 to 1100 MeV. 10 500 10 550 10 600 10 650 10 700 10 750 10 8007072747678 s (cid:64) MeV (cid:68) (cid:200) T FIG. 20. | T | as a function of the √ s center of mass energy when only the B ∗ ¯ B ∗ channel isconsidered ( J = 1 case). Each curve is related to the cut off values q max equal to 700 , , , In Fig. 20 we show the plot for | T | as a function of the center of mass energy of thesystem. Note that in this case, we also have a peak about 10650 MeV, which is just thethreshold mass of the B ∗ ¯ B ∗ channel. Again, we see essentially a cusp in the amplitude whichdoes not correspond to a bound state. The situation is similar if we increase the value of˜ t B ∗ ¯ B ∗ → B ∗ ¯ B ∗ of a factor 1 . | T | growsaccordingly, but the cusp remains and its shape is like in Fig. 20. V. SUMMARY AND CONCLUSION Using the local hidden gauge lagrangians, we have studied the B ¯ B ∗ and B ∗ ¯ B ∗ interactionsfor isospin I = 1. We show that the exchange of a light meson is not allowed by OZI rule.20or that reason we have investigated the contributions coming from heavy vector exchangeand also due to the two pion exchange, interacting and noninteracting among themselves,in which the OZI restriction no longer holds. Unlike Refs. [38, 39], the vector exchangepotential is not the main source of the interactions here. In view of this, we consider thevector exchange potential corrected by a factor that takes into account the contributions ofthe others mesons exchange cases and, then we use it as the kernel of the Bethe-Salpeterequation in order to solve the transition matrix T . Looking for poles in the T matrix, wetried to relate them with the Z b (10610) and Z b (10650) states reported by Belle collaboration.From our results, using a cut off value q max = 700 MeV, we found a bound state of B ¯ B ∗ withmass in the range 10587 − Z b (10610)at 10608 MeV. In the case of B ∗ ¯ B ∗ interaction, we found a cusp at 10650 MeV for spin J = 0 and J = 2 cases. On the other hand, the spin J = 1 case can be considered only inthe one channel problem without taking into account the ρ Υ channel. In this case, again acusp at 10650 MeV appears in the | T | as can be seen in Fig. 20 and was also pointed outin Ref. [13, 14]. 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