The bottomed strange molecules with isospin 0
aa r X i v : . [ h e p - ph ] J a n The bottomed strange molecules with isospin 0
Zhi-Feng Sun , ∗ Ju-Jun Xie , † and E. Oset ‡ School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Departamento de Fsica Terica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigac ´ i on de Paterna, Aptdo. 22085, 46071 Valencia, Spain (Dated: January 16, 2018)Using the local hidden gauge approach, we study the possibility of the existence of bottomedstrange molecular states with isospin 0. We find three bound states with spin-parity 0 + , 1 + and 2 + generated by the ¯ K ∗ B ∗ and ωB ∗ s interaction, among which the state with spin 2 can be identifiedas B ∗ s (5840). In addition, we also study the ¯ K ∗ B and ωB s interaction and find a bound statewhich can be associated to B s (5830). Besides, the ¯ KB ∗ and ηB ∗ s and ¯ KB and ηB s systems arestudied, and two bound states are predicted. We expect that further experiments can confirm ourpredictions. PACS numbers: 14.40.Be, 13.25.Jx, 12.38.Lg
I. INTRODUCTION
The local hidden gauge symmetry was introduced in Refs. [1–4] which regards vector mesons as the gauge bosons andpseudoscalar mesons as the Goldstone bosons. Considering this symmetry together with the global chiral symmetry,one can construct the Lagrangian describing interactions involving vector and pseudoscalar mesons. On the otherhand, the Bethe-Salpeter equation is a powerful tool to deal with nonperturbative physics restoring two body unitarityin coupled channels. The theory incorporating the above two points has been instrumental in explaining manyproperties of hadronic resonances. In Ref. [5], the f (1370) and f (1270) are explained as resonances generatedfrom ρρ interaction. Later, in Ref. [6] the work of [5] was extended to SU(3), and five of the generated states canbe identified with the observed f (1370), f (1270), f (1710), f ′ (1525), K ∗ (1430). In the spin 1 sector, a resonancewas also found in Ref. [6] with mass and width around 1800 and 80 MeV, respectively. This state, h (1800), isdynamically generated from the K ∗ ¯ K ∗ interaction, and it was investigated in the J/ψ → ηK ∗ ¯ K ∗ in Ref. [7] andin the η c → φK ∗ ¯ K ∗ in Ref. [8]. In Ref. [9], the authors studied the interactions of ρ , ω and D ∗ , and three stateswith spin J = 0 , , D ∗ (2640) and D ∗ (2460), respectively. The third state predicted, D (2600), was found later by [10] and has been reconfirmed [11, 12].This work was extended to the case of ρ ( ω ) B ∗ ( B ) interaction in Ref. [13], where B (5721) and B ∗ (5747) are explainedas ρ ( ω ) B ∗ and ρB molecules.First evidence for at least one of the bottomed strange states was found by the OPAL experiment [14]. Evidencefor a single state interpreted as B ∗ s was seen by the Delphi Collaboration [15]. B ∗ s (5840) was observed by both CDFand D0 in the B + K − channel [16–18]. In the CDF experiment, there is another peak in the B + K − invariant massspectrum corresponding to B s (5830). However, B s (5830) → B + K − is not allowed. The interpretation is that thispeak comes from the channel B ∗ + K − and B ∗ + decays to B + γ where the photon is not detected. As a consequence,the peak is shifted by B ∗ − B mass difference due to the missing momentum of the photon. Recently, LHCb firstmeasured the mass and width of B ∗ s (5840) in the B ∗ + K − channel. Besides, the ratio B ∗ s (5840) → B ∗ + K − B ∗ s (5840) → B + K − was alsomeasured and the decay of B s (5830) → B ∗ + K − was observed as well [19].In this work, we extrapolate the local hidden gauge approach to the systems containing bottomed and strangequarks. The paper is organized as follows. After this introduction, in section II we will show the local hidden gaugeLagrangian, from which the potentials are obtained. And then we construct the T matrix by solving the Bethe-Salpeterequation. In section III, the results are given. Finally, we make a short summary. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: Eulogio.Oset@ific.uv.es
II. FORMALISMA. Lagrangian
In order to describe the interaction of bottomed and strange mesons, we need to use the local hidden gauge approach,under which vector mesons are treated as gauge bosons. The covariant derivative is defined as D µ ξ L,R = ∂ µ ξ L,R − iV µ ξ L,R , (1)and the gauge field strength as V µν = ∂ µ V ν − ∂ ν V µ − ig [ V µ , V ν ] . (2)Here, g is given by g = m V f π with the pion decay constant f π = 93 MeV, and m V the mass of vector mesons. ξ L,R isdefined as ξ L = e iσ/f σ e − i √ P/f π , (3) ξ R = e iσ/f σ e i √ P/f π . (4)In this paper, we take the unitary gauge, i.e., σ = 0. In the above equations, the matrices V µ and P have the followingform V µ = ω √ + ρ √ ρ + K ∗ + B ∗ + ρ − ω √ − ρ √ K ∗ B ∗ K ∗− ¯ K ∗ φ B ∗ s B ∗− ¯ B ∗ ¯ B ∗ s Υ µ ,P = η √ + η ′ √ + π √ π + K + B + π − η √ + η ′ √ − π √ K B K − ¯ K − η √ + q η ′ B s B − ¯ B ¯ B s η b . (5) After defining the blocks ˆ α ⊥ µ = 12 i (cid:16) D µ ξ R · ξ † R − D µ ξ L · ξ † L (cid:17) , ˆ α k µ = 12 i (cid:16) D µ ξ R · ξ † R + D µ ξ L · ξ † L (cid:17) , (6)one can construct the Lagrangian [4] L = L A + a L V + L III , (7)where L A = f π h ˆ α ⊥ µ ˆ α µ ⊥ i ,a L V = f σ h ˆ α k µ ˆ α µ k i , L III = − h V µν V µν i , (8)with f σ = af π , and we take a = 2 as in Ref. [4].After expanding the Lagrangians in Eq. (7), we get the terms needed in our calculation, i.e., three vector vertex L V V V = ig h ( ∂ µ V ν − ∂ ν V µ ) V µ V ν i , (9)four vector vertex L V V V V = g h V µ V ν V µ V ν − V ν V µ V µ V ν i , (10)four pseudoscalar vertex L P P P P = − f π h [ P, ∂ µ P ][ P, ∂ µ P ] i (11)and vector pseudoscalar pseudoscalar vertex L V P P = − ig h V µ [ P, ∂ µ P ] i . (12)Note that there is no V V P P contact term under the hidden local symmetry. Moreover, since the VVP interaction isanomalous with a comparatively small contribution, we do not take it into account. In this work, we will study theinteraction between bottom and strange mesons, so we extend the SU(3) flavor symmetry to SU(4). Next we changethe form of the three vector Lagrangian in Eq. (9) through some short calculations L = ig h ( ∂ µ V ν − ∂ ν V µ ) V µ V ν i = ig h ∂ µ V ν V µ V ν − ∂ ν V µ V µ V ν i = ig h ∂ ν V µ V ν V µ − ∂ ν V µ V µ V ν i = ig h V µ ∂ ν V µ V ν − ∂ ν V µ V µ V ν i = ig h [ V µ , ∂ ν V µ ] V ν i = ig h V µ [ V ν , ∂ µ V ν ] i , (13)from which we see that this Lagrangian has a similar form as that in Eq. (12) except for the minus sign.As noted in [24], for small three momenta of the vector mesons compared to their mass, the ǫ component of theexternal vectors can be neglected. Then V µ in the last of Eq. (13) should be V i (i=1, 2, 3) if it corresponds to anexternal vector, but then ∂ i will give a three momentum of this vector, which one is neglecting. Hence V µ cannotcorrespond to the external vectors and is necessarily the exchanged vector. The rest of the operator V ν V ν gives riseto ǫ µ ǫ µ → − ~ǫ · ~ǫ and the last of the Eq. (13) is equivalent to Eq. (12) including the sign.It should be noted that the local hidden gauge approach is constructed within SU(2) or SU(3) [25, 26]. In the heavyquark sector one cannot invoke heavy mesons as Goldstone bosons. Yet, the extension to the heavy quark sector ispossible because the dominant terms of the interaction correspond to the exchange of light vectors, ρ, ω, φ and theheavy quarks of the hadrons are just spectators. In this case it is possible to make a mapping of the interaction in theheavy light hadron sector to the one in the heavy hadron sector. For practical purposes one can use the local hiddengauge Lagrangians extrapolated to SU(4) as in Eq. (5), since for the exchange of light vectors one is only making useof the relevant SU(3) subgroup. Discussion on this issue and the proof of this property can be seen in section II of[21] and section II and Appendix of [27]. B. B ∗ and ¯ K ∗ interaction The interaction terms of ¯ K ∗ B ∗ and ωB ∗ s are depicted by the diagrams in Fig. 1, including contact terms and t-channel diagrams. Here, we neglect the bottomed-meson-exchange diagrams, which have a much smaller contributiondue to the heavy mass of bottomed mesons. Besides, the amplitude of ωB ∗ s → ωB ∗ s is zero, because of the OZI (Okubo-Zweig-Iizuka) rule [28–30]. Recalling the isospin doublet ( K ∗ + , K ∗ ), ( ¯ K ∗ , − K ∗− ), ( B ∗ + , B ∗ ), ( ¯ B ∗ , − B ∗− ), andthe isospin triplet ( − ρ + , ρ , ρ − ), we have the flavor wave functions | ¯ K ∗ B ∗ ; I = 0 i = K ∗− B ∗ + + ¯ K ∗ B ∗ √ , (14) | ωB ∗ s ; I = 0 i = ωB ∗ s . (15)Here the channel φB ∗ s is not considered, since its threshold is much higher than the other two. With the structureof Eqs. (12) and (13), all the amplitudes have the structure of ( k + k ) · ( k + k ) ǫ µ ǫ µ ǫ ν ǫ ν . After writing theamplitudes using Feynman rules, we project the polarization vector products into different spin states: P (0) = 13 ǫ µ ǫ µ ǫ ν ǫ ν , (16) P (1) = 12 ( ǫ µ ǫ ν ǫ µ ǫ ν − ǫ µ ǫ ν ǫ ν ǫ µ ) , (17) P (2) = 12 ( ǫ µ ǫ ν ǫ µ ǫ ν + ǫ µ ǫ ν ǫ ν ǫ µ ) − ǫ µ ǫ µ ǫ ν ǫ ν (18) FIG. 1: Feynman diagrams describing ¯ K ∗ B ∗ and ωB ∗ s interaction. with the order of the ǫ ’s as 1 , , , → K ∗ B ∗ → ¯ K ∗ B ∗ with I = 0 as follows: t S =0 cont = 4 g , (19) t S =1 cont = 6 g , (20) t S =2 cont = − g , (21) t S =0 , , ex = − g (cid:18) m ρ + 1 m ω (cid:19) ( s − u ) , (22)for ¯ K ∗ B ∗ → ωB ∗ s t S =0 cont = − g , (23) t S =1 cont = 0 , (24) t S =2 cont = 2 g , (25) t S =0 , , ex = g m K ∗ ( s − u ) . (26)In the above equations, the Mandelstam variables s and u are defined as s = ( k + k ) , (27) u = ( k − k ) . (28) C. B and ¯ K ∗ and B ∗ and ¯ K interactions In Fig. 2, we show the diagrams for the ¯ K ∗ B and ωB s interaction. Note that under hidden local symmetry, thereis no contact term for vector pseudoscalar scattering. The amplitude of ωB s → ωB s is zero, because of the OZI rules.For ¯ K ∗ B → ¯ K ∗ B in I = 0 we need the exchange of ρ and ω and we obtain t S =1 ex = − g (cid:18) m ρ + 1 m ω (cid:19) ( s − u ) (29)and for ¯ K ∗ B → ωB s t S =1 ex = g m K ∗ ( s − u ) . (30) FIG. 2: Feynman diagrams describing ¯ K ∗ B and ωB s interaction.FIG. 3: Feynman diagrams describing ¯ KB ∗ and ηB ∗ s interaction. Similarly, we can also get the amplitudes for the ¯ KB ∗ → ¯ KB ∗ process in I = 0 as follows t S =1 ex = − g (cid:18) m ρ + 1 m ω (cid:19) ( s − u ) . (31)However, according to the diagrams shown in Fig. 3, the calculation for ¯ KB ∗ → ηB ∗ s in I = 0 is a little bit different.Using Feynman rule and considering the flavor wave function, we obtain t S =1 ex = − √ g m K ∗ ( s − u ) . (32) D. B and ¯ K interaction In Fig. 4, we show the diagrams depicting the interaction of pseudoscalar and pseudoscalar mesons. The amplitudeof contact terms corresponding to Eq. (11) are obtained for ¯ KB → ¯ KB process in I = 0 as t S =0 cont = − f (2 u − t − s ) , (33)for ¯ KB → ηB s process t S =0 cont = − √ f ( s − u ) (34)and for ηB s → ηB s process t S =0 cont = − f ( − t + u + s ) (35)with t = ( k − k ) . The amplitude of t-channel diagrams for ¯ KB → ¯ KB have the following expressions t S =0 ex = − g (cid:18) m ρ + 1 m ω (cid:19) ( s − u ) , (36) FIG. 4: Feynman diagrams describing ¯ KB and ηB s interaction. and for ¯ KB → ηB s t S =1 ex = − √ g m K ∗ ( s − u ) . (37)The t channel diagrams for ηB s → ηB s has 0 contribution. E. T-matrix
With the preparation above, using the Bethe-Salpeter equation in its on-shell factorized form, we obtain the T-matrix T = ( I − V G ) − V, (38)where V corresponds to the transition amplitudes shown above, but projected to s -wave. So we neglect the product ~k · ~k in the Mandelstam variables u and t which corresponds to p -wave contribution, i.e., u ≈ m + m + m + m − ( m − m )( m − m )2 s ,t ≈ m + m + m + m m − m )( m − m )2 s . (39) G is the two-meson loop function G = i Z d q (2 π ) q − m + iǫ P − q ) − m + iǫ . (40)Using a cut off of the three momentum, we have G = Z q max q dq (2 π ) ω + ω ω ω [( P ) − ( ω + ω ) + iǫ ] . (41) FIG. 5: The vertex of B ∗ B ∗ ρ and K ∗ K ∗ ρ at the hadronic level. This integral was already done (see Ref. [31]), and we show it as follows G = 132 π νs log s − ∆ + ν q m q max − s + ∆ + ν q m q max + log s + ∆ + ν q m q max − s − ∆ + ν q m q max − ∆ s log m m +2 ∆ s log 1 + q m q max q m q max + log m m q max − " s m q max ! s m q max ! . (42)In Eqs. (40), (41) and (42), P is the total four-momentum of the two mesons in the loop, m and m are the masses, q max stands for the cut off, ω i = p ~q i + m i , P is nothing but the center-of-mass energy √ s , ∆ = m − m , and ν = p [ s − ( m + m ) ][ s − ( m − m ) ]. III. RESULTSA. Discussion of the couplings under SU(4) symmetry
In this subsection, we follow Refs. [13, 20, 21] and discuss the couplings in the Lagrangian. As an example, weconsider the vertex of B ∗ B ∗ ρ . In order to estimate the corresponding coupling, we need to compare this vertex withthat of K ∗ K ∗ ρ , since their topology is the same if the ¯ s and ¯ b quarks are seen as spectators. Fig. 5 shows the diagramsfor these two vertices at the quark level, in which case the corresponding S matrices should be the same, i.e., S mic = 1 − it r m L E L s m ′ L E ′ L s ω ρ V / (2 π ) δ ( P in − P out ) . (43)On the other hand, at the hadronic level, the S matrices are written as S macB ∗ = 1 − it B ∗ √ ω B ∗ √ ω B ∗ p ω ρ V / (2 π ) δ ( P in − P out ) , (44) S macK ∗ = 1 − it K ∗ √ ω K ∗ √ ω K ∗ p ω ρ V / (2 π ) δ ( P in − P out ) . (45)As discussed above, we should have S macB ∗ = S macK ∗ which tells us that the corresponding T matrices obey the relationat the threshold as follows t B ∗ t K ∗ = m B ∗ m K ∗ . (46)If we use the Lagrangian in Eq. (9) and calculate the T matrices of the processes in Fig. 5, we find that Eq. (46)holds automatically, when the ρ is the exchanged (virtual) vector meson, because the amplitude has the ∂ µ ∼ = ∂ operator acting on the external vectors. The coupling of B ∗ B ∗ ρ in Eq. (9) implements correctly the field correctionfactor of Eq. (46). Since in this case the b quark acts as a spectator in the vertex, automatically this amplitude isconsistent with heavy quark spin symmetry [22]. Similar discussions can be applied to the BBρ vertex with respectto
KKρ , and we have t B t K = m B m K , (47)but this is what we obtain from Eq. (12) using SU(4) flavor symmetry. Effectively one is using SU(3) when the heavyquark is considered as a spectator. In summary, we apply the Lagrangians of section II-A, and this takes automaticallyinto account all the elements discussed above. B. The ¯ K ∗ B ∗ system With the potentials given in above section, we solve the Bethe-Salpeter equation considering ¯ K ∗ B ∗ , ωB ∗ s and φB ∗ s coupled channels. And we obtain three bound states with J = 0 , ,
2, using the cutoff q max around 1055 ∼ . ∼ . B ∗ s (5840). With this q max , we predict that the bound state with J = 0 has a mass 5908 . ∼ . J = 1 has amass of 5912 . ∼ . B s (5830) with spin 1 is smaller than that of B ∗ s (5840). However, the generated bound statewith spin 1 has a mass about 65 MeV larger than that of the bound state with spin 2. Henceforth, it is difficult toexplain the B s (5830) as the ¯ K ∗ B ∗ bound state. In the next subsection, we will come back to this problem.The T-matrix close to a pole behaves like T ij ≈ g i g j z − z R , (48)where i, j = ¯ K ∗ B ∗ , ωB ∗ s , φB ∗ s , g i is the coupling to the channel i , Re ( z R ) gives the mass of the bound state, Im ( z R )the half width, and z is the complex value of the Mandelstam variable s . The coupling for a certain channel is obtainedas g i = lim z → z R T ii ( z − z R ) . (49)The sign of the coupling to the B ∗ ¯ K ∗ channel is chosen as positive, and those for the other channels are thendetermined by the following formula g i g j = lim z → z R T ii T ij . (50) TABLE I: The couplings for ¯ K ∗ B ∗ systems mixing with ωB ∗ s , φB ∗ s channels. Here we chose the typical value of the cut off as1070 MeV. All the values are given in units of MeV.channel J=0 J=1 J=2¯ K ∗ B ∗ ωB ∗ s -10696 -14810 -15017 φB ∗ s The value of the couplings are listed in Tab. I, from which we can see that the ¯ K ∗ B ∗ component is dominant forall the states. C. The ¯ K ∗ B system As mentioned in the previous subsection, the B s (5830) can not be explained as ¯ K ∗ B ∗ bound state with spin 1,since in PDG the mass of B s (5830) is smaller than that of B ∗ s (5840), which is contrary to our results. Now whatwe do is trying to explain the B s (5830) under the ¯ K ∗ B/ωB s system.Under hidden local symmetry there are no contact terms for V V P P vertex, so that only vector exchange diagramsare involved. For the vector exchange terms, the interactions we study in this subsection have the same form as thatof the ¯ K ∗ B/ωB s /φB s interactions. So here we expect to find a bound state like in the case of the ¯ K ∗ B ∗ system. Weuse q max = 1055 ∼ K ∗ B ∗ bound state with spin 2. Then we obtain a pole position inthe range of 5822 . ∼ . B s (5830) in the PDG. In Fig. 7, we plot theline shape of the | T | depending on the center-of-mass energy √ s . We also calculate the couplings, which have thevalue of g ¯ K ∗ B = 47654, g ωB s = − g φB s = 18855 with the cut off q max = 1070 MeV. max =1055 MeVq max =1070 MeV | T | / s (MeV)B*K* (J=0) q max =1085 MeV _ B*K* (J=1) q max =1055 MeVq max =1070 MeVq max =1085 MeV | T | / s (MeV)_ max =1055 MeVq max =1070 MeVq max =1085 MeV B*K* (J=2) | T | / s (MeV)_ FIG. 6: Squared amplitude for ¯ K ∗ B ∗ /ωB ∗ s /φB ∗ s systems with spin 0, 1 and 2, respectively. max =1055 MeVq max =1070 MeVq max =1085 MeV _ BK* | T | / s (MeV)_ FIG. 7: Squared amplitude for ¯ K ∗ B/ωB s /φB s sector depending on the center-of-mass energy. D. Other predictions
In this subsection, we will show the results corresponding to ¯ KB ∗ /ηB ∗ s and ¯ KB/ηB s interactions.Like the case of ¯ K ∗ B/ωB s /φB s system, there are no contact terms for ¯ KB ∗ /ηB ∗ s interaction. Only the vectormeson exchange diagrams are considered. In Fig. 8, we plot the squared amplitude depending on the center-of-massenergy √ s . Here, we also use the cut off q max = 1055 ∼ . ∼ . B ∗ ¯ K and B ∗ s η are 30637 MeV and − max =1055 MeVq max =1070 MeVq max =1085 MeV BK | T | / s (MeV)_ max =1055 MeVq max =1070 MeVq max =1085 MeV B*K | T | / s (MeV)_ FIG. 8: Squared amplitude for ¯
KB/ηB s and ¯ KB ∗ /ηB ∗ s sector.TABLE II: Summary of our results where the cut off is in the range of 1055 ∼ I ( J p ) Main component Exp. State mass I ( J p ) Main component Exp.5475 . ∼ . + ) ¯ KB - 5908 . ∼ . + ) ¯ K ∗ B ∗ -5671 . ∼ . + ) ¯ KB ∗ - 5912 . ∼ . + ) ¯ K ∗ B ∗ -5822 . ∼ . + ) ¯ K ∗ B B s (5830) 5847 . ∼ . + ) ¯ K ∗ B ∗ B ∗ s (5840) the cut off as 1070 MeV.For ¯ KB/ηB s system, we predict a bound state with a mass of 5475 . ∼ . g ¯ KB = 53577MeV and g ηB s = − q max = 1070 MeV.In TABLE. II, we list our results of all the systems. IV. SUMMARY
In this work, we have studied the systems containing bottomed and strange quarks by the chiral unitary approach.Considering ¯ K ∗ B ∗ and ωB ∗ s coupled channels and solving the Bethe-Salpeter equation, we find three states withmasses 5908 . ∼ . . ∼ . . ∼ . q max chosen as1055 ∼ B ∗ s (5840). From the couplings that we obtained, wecan see that the ¯ K ∗ B ∗ component is dominant. However, the B s (5830) can not be explained as the state with spin1, since its mass is smaller than that of B ∗ s (5840). So we studied another system, i.e., ¯ K ∗ B/ωB s system, and weget a bound state with a mass 5822 . ∼ . B s (5830). In addition, we alsostudied ¯ KB ∗ /ηB ∗ s and ¯ KB/ηB s interactions, and predict two bound states with masses 5671 . ∼ . . ∼ . Acknowledgments
This work is partly supported by the National Science Foundation for Young Scientists of China under GrantsNO. 11705069 and the Fundamental Research Funds for the Central Universities. It is partly supported by theNational Natural Science Foundation of China (Grants No. 11475227, 11735003) and the Youth Innovation PromotionAssociation CAS (No. 2016367). This work is also partly supported by the Spanish Ministerio de Economia yCompetitividad and European FEDER funds under the contract number FIS2011-28853-C02-01, FIS2011- 28853-C02-02, FIS2014-57026-REDT, FIS2014-51948-C2- 1-P, and FIS2014-51948-C2-2-P, and the Generalitat Valencianain the program Prometeo II-2014/068. [1] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. , 1215 (1985).[2] M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. , 217 (1988).[3] U. G. Meissner, Phys. Rept. , 213 (1988).[4] M. Harada and K. Yamawaki, Phys. Rept. , 1 (2003)[5] R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D , 114018 (2008)[6] L. S. Geng and E. Oset, Phys. Rev. D , 074009 (2009)[7] J. J. Xie, M. Albaladejo and E. Oset, Phys. Lett. B , 319 (2014)[8] X. L. Ren, L. S. Geng, E. Oset and J. Meng, Eur. Phys. J. A , 133 (2014)[9] R. Molina, H. Nagahiro, A. Hosaka and E. Oset, Phys. Rev. D , 014025 (2009)[10] P. del Amo Sanchez et al. [BaBar Collaboration], Phys. Rev. D , 111101 (2010)[11] R. Aaij et al. [LHCb Collaboration], JHEP , 145 (2013)[12] R. Aaij et al. [LHCb Collaboration], Phys. Rev. D , no. 7, 072001 (2016)[13] P. Fernandez-Soler, Z. F. Sun, J. Nieves and E. Oset, Eur. Phys. J. C , no. 2, 82 (2016)[14] R. Akers et al. [OPAL Collaboration], Z. Phys. C , 19 (1995).[15] M. Moch [DELPHI Collaboration], PoS HEP , 232 (2006).[16] R. K. Mommsen, Nucl. Phys. Proc. Suppl. , 172 (2007)[17] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. , 082001 (2008)[18] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. , 082002 (2008)[19] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , no. 15, 151803 (2013)[20] W. H. Liang, C. W. Xiao and E. Oset, Phys. Rev. D , no. 5, 054023 (2014)[21] S. Sakai, L. Roca and E. Oset, Phys. Rev. D , no. 5, 054023 (2017)[22] A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000)[23] C. Patrignani et al. [Particle Data Group], Chin. Phys. C , no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001[24] E. Oset and A. Ramos, Eur. Phys. J. A , 445 (2010)[25] G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B , 425 (1989).[26] H. Nagahiro, L. Roca, A. Hosaka and E. Oset, Phys. Rev. D , 014015 (2009)[27] W. H. Liang, J. M. Dias, V. R. Debastiani and E. Oset, arXiv:1711.10623 [hep-ph].[28] S. Okubo, Phys. Lett. , 165 (1963).[29] G. Zweig, Developments in the Quark Theory of Hadrons, Volume 1. Edited by D. Lichtenberg and S. Rosen. pp. 22-101[30] J. Iizuka, Prog. Theor. Phys. Suppl. , 21 (1966).[31] J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. D , 074001 (1999) Erratum: [Phys. Rev. D , 099906 (1999)] Erratum:[Phys. Rev. D75