Exploring the ϒ(6S) χ bj φ and ϒ(6S) χ bj ω hidden-bottom hadronic transitions

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1 Eur. Phys. J. C (07) 77:65 DOI 0.40/epjc/s Regular Article - Theoretical Physics Exploring the ϒ(6S) χ bj φ and ϒ(6S) χ bj ω hidden-bottom hadronic transitions Qi Huang,,a, o Wang,,b, Xiang Liu,,c, Dian-Yong Chen 3,d, Takayuki Matsuki 4,5,e School of Physical Science and Technology, Lanzhou University, Lanzhou , China Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou , China 3 Department of Physics, Southeast University, Nanjing 0094, China 4 Tokyo Kasei University, -8- Kaga, Itabashi, Tokyo , Japan 5 Theoretical Research Division, Nishina Center, RIKEN, Wako, Saitama , Japan Received: 5 January 07 / Accepted: 7 February 07 / Published online: 7 March 07 The Author 07. This article is an open access publication Abstract In this work, we investigate the hadronic loop contributions to the ϒ(6S) χ bj φ(j = 0,, ) along with ϒ(6S) χ bj ω(j = 0,, ) transitions. We predict that the branching ratios of ϒ(6S) χ b0 φ, ϒ(6S) χ b φ and ϒ(6S) χ φ are ( ) 0 6, ( ) 0 6, and (. 5.8) 0 6, respectively, and those of ϒ(6S) χ b0 ω, ϒ(6S) χ b ω and ϒ(6S) χ ω are (0.5.8) 0 3, ( ) 0 3, and ( ) 0 3, respectively. Especially, some typical ratios, which reflect the relative magnitudes of the predicted branching ratios, are given, i.e., for ϒ(6S) χ bj φ transitions, R φ 0 = [ϒ(6S) χ b φ/[ϒ(6s) χ b0 φ 0.74, R φ 0 = [ϒ(6S) χ φ/[ϒ(6s) χ b0 φ 3.8, and R φ = [ϒ(6S) χ φ/[ϒ(6s) χ b φ 4.43, and for ϒ(6S) χ bj ω transitions, R0 ω = [ϒ(6S) χ bω/ [ϒ(6S) χ b0 ω 4., R0 ω = [ϒ(6S) χ ω/ [ϒ(6S) χ b0 ω 7.06, and R ω = [ϒ(6S) χ ω/ [ϒ(6S) χ b ω.7. With the running of elleii in the near future, experimental measurement of these two kinds of transitions will be a potential research issue. Introduction As an interesting research issue, experimental studies of the hadronic transitions of ϒ(5S) have been focused on by the elle Collaboration in the past decade. When surveying the reported hadronic transitions of ϒ(5S), we found their gena huangq6@lzu.edu.cn b wangb3@lzu.edu.cn c xiangliu@lzu.edu.cn d chendy@seu.edu.cn e matsuki@tokyo-kasei.ac.jp eral property, i.e., their observed hadronic transitions have large branching ratios. For example, elle observed anomalous decay widths of the ϒ(5S) ϒ(nS)π + π [, and ϒ(5S) χ bj ω(j = 0,, ) transitions [. In addition, the two bottomonium-like states Z b (060) and Z b (0650) were observed in ϒ(5S) ϒ(nS)π + π [3. As indicated in a series of theoretical studies [4 0, the puzzling phenomena occurring in ϒ(5S) transitions reflect an underlying mechanism mediated by a coupled-channel effect, since ϒ(5S) is above the thresholds of () () [. In the bottomonium family, the ϒ(6S) is in a similar situation to that of ϒ(5S). We have reason to believe that the coupled-channel effect is still important to the hadronic transitions of ϒ(6S), whose exploration is, thus, an intriguing topic. This theme can provide us a valuable information of the coupled-channel effect on these decays. In this work, we calculate the ϒ(6S) χ bj φ (J = 0,, ) along with ϒ(6S) χ bj ω (J = 0,, ) processes via, which is an equivalent description of the coupled-channel effect [4 6,8 0, 6. y analyzing these transitions, the relative decay rates of ϒ(6S) χ bj φ (J = 0,, ) and ϒ(6S) χ bj ω (J = 0,, ), which are a typical physical quantity given by our calculation, are determined. Especially, our results show that these relative decay rates are weakly dependent on the model parameters. Thus, experimental measurement of these rates can be a crucial test of in the ϒ(6S) χ bj φ and ϒ(6S) χ bj ω decays. In addition, we also estimate the typical values of the branching ratios of ϒ(6S) χ bj φ and ϒ(6S) χ bj ω, which can be measured experimentally in the near future. Anyway, we would like to inspire experimentalists interest in searching for the ϒ(6S) χ bj φ and ϒ(6S) χ bj ω decays by the results presented in this work. 3

2 65 Page of 0 Eur. Phys. J. C (07) 77 :65 This paper is organized as follows. After the introduction, we present the detailed calculation of ϒ(6S) χ bj φ and ϒ(6S) χ bj ω via in Sect.. The numerical results are presented in Sect. 3. The paper ends with a short summary. ϒ(6S) χ bj φ and ϒ(6S) χ bj ω transitions via hadronic loop mechanism Under, the ϒ(6S) χ bj φ transitions occur via the triangle loops composed of s ()0 and s ()0, which play a role of the bridge to connect the initial state ϒ(6S) and final states φ and χ bj.infigs.,, and 3, we list the typical diagrams depicting the ϒ(6S) χ bj φ(j = 0,, ) transitions. For the ϒ(6S) χ bj ω transitions, due to very different quark contents between φ and ω, the bridges change to () and () and the diagrams change simultaneously as in Figs. 4, 5, and 6. To calculate these diagrams at the hadron level, we adopt the effective Lagrangian approach, in which we first introduce the Lagrangians relevant to our calculation. For the interactions between a heavy quarkonium and two heavy light mesons, the Lagrangians are constructed based on the heavy quark effective theory. In the heavy quark limit, the light degrees of freedom s l is a good quantum number. Thus, each value of s l is assigned to a doublet formed by the states with a total angular momentum J = s l ± /, while for the heavy quarkonium, since the degeneracy is expected under the rotations of two heavy quark spins, there is a multiplet formed by heavy quarkonia with the same angular momentum l. Therefore, under the framework of heavy quark symmetry, general forms of couplings between an S-wave or P-wave heavy quarkonium and two heavy light mesons can be constructed as [7 Fig. Schematic diagrams depicting the ϒ(6S) χ b φ process via Fig. 3 Schematic diagrams depicting the ϒ(6S) χ φ process via Fig. Schematic diagrams depicting the ϒ(6S) χ b0 φ process via Fig. 4 Schematic diagrams depicting the ϒ(6S) χ b0 ω process via 3

3 Eur. Phys. J. C (07) 77 :65 Page 3 of 0 65 P (Q Q) μ = + /v [ χ μα γ α + ε μαβγ v α γ β χ bγ + ( γ μ v μ) χ b0 + h μ /v b γ 5 3, (3) respectively. H (Q q) represents a doublet formed by heavy light pseudoscalar and vector mesons [7 0 H (Q q) = + /v [ μ γ μ γ 5, (4) Fig. 5 Schematic diagrams depicting the ϒ(6S) χ b ω process via with definitions () = ( ()+, ()0, s ()0 ) and () = ( (), ()0, s ()0 ) T as in Ref. [3. H ( Qq) corresponds to a doublet formed by heavy light anti-mesons, which can be obtained by applying the charge conjugation operation to H (Q q). For the interaction between a light vector meson and two heavy light mesons, the general form of the Lagrangian reads [7, 5 L V = iβtr[h j v μ ( ρ μ ) i j H i +iλtr[h j σ μν F μν (ρ) H i, (5) where ρ μ = i g V V μ, (6) F μν (ρ) = μ ρ ν ν ρ μ +[ρ μ,ρ ν, (7) and the vector octet V has the form (ρ 0 + ω) ρ + K + V = ρ ( ρ 0 + ω) K 0. (8) K K 0 φ Fig. 6 Schematic diagrams depicting the ϒ(6S) χ ω process via L s = igtr [R (Q Q) H ( Qq) γ μ μ H (Q q) + H.c., [ L p = ig Tr P (Q Q)μ H ( Qq) γ μ H (Q q) + H.c., () y expanding the Lagrangians in Eqs. () and (5), the following concrete expressions are obtained: L ϒ () () = ig ϒϒ μ ( μ μ ) + g ϒ ε μναβ μ ϒ ν ( α β β α ) + ig ϒ ϒμ ( ν ν μ ν μ ν ν μ ν ), (9) in which R (Q Q) and P (Q Q) denote multiplets formed by bottomonia with l = 0 and l =, and their detailed expressions, as in Ref. [8, can be written as R (Q Q) = + /v [ ϒ μ γ μ η b γ 5 /v, () L χbj () () = g χ b0 χ b0 g χb0 χ b0 μ μ + ig χc χμ b ( μ μ ) g χ χ μν μ ν + g χ χμν ( μ ν + ν μ ) ig χ ε μναβ α χ μρ ( ρ ν β β ρ ν ), (0) 3

4 65 Page 4 of 0 Eur. Phys. J. C (07) 77 :65 L () () V = ig V i μ j (V μ ) i j f Vε μναβ ( μ V ν ) i j ( i α β i j ) + ig V ν i μ j ν (V μ) i j α β j + 4if V iμ ( μ V ν ν V μ ) i j ν j. () With the above effective Lagrangians, we can write out the amplitudes of hadronic loop contributions to ϒ(6S) χ bj φ(j = 0,, ). Fortheϒ(6S) χ b0 φ transition, the amplitudes corresponding to Fig. are M (0 ) = (π) 4 [ ig ϒ s s ɛ μ ϒ ((ik ) μ (ik ) μ ) [ ig s s φɛφλ (( ik ) λ (iq) λ )[ g s s χ b0 k m s k m s q m F (q ), () s M (0 ) = (π) 4 [ g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f s s φε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ )[ g s s χ b0 gσ α + k αk σ /m s k m s k m s q m F (q ), (3) s M (0 3) = (π) 4 [g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f s s φ ε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) ζ [ g s s χ gα + k α k ζ /m s b0 k m s k m s M (0 4) = gσ ζ + qσ q ζ /m s q m s F (q ), (4) (π) 4 [ig ϒ s s ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ɛφλ (ig s s φ g δσ (( ik ) λ (iq) λ ) + 4if s s φ (ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ g s s χ b0 g σζ + q σ q ζ /m s q m F (q ), (5) s where p, p, and p 3 are momenta of ϒ(6S), φ/ω and χ bj, and k, k, and q are momenta of internal () and exchanged (), respectively. In these expressions of the decay amplitudes, the monopole form factor is introduced, by which the inner structure of the interaction vertices is reflected and the off-shell effect of the exchanged bottom strange mesons is compensated. Here, the adopted form factor is taken as F (q ) = (m E )/(q ), with m E being the mass of the exchanged boson and the cutoff being parameterized as = m E + α QCD with QCD = 0. GeV as in Refs. [4 6. We need to specify that the monopole behavior of the adopted form factor was suggested by the QCD sum rule studied in Ref. [6. In a series of published papers (see Refs. [4 6,8 0, 6,8,7), the monopole form factor was adopted to study the transitions of charmonia and bottomonia, and decays. Thus, this approach has been tested by these successful studies. In a similar way, we can further write out the decay amplitudes of ϒ(6S) χ b φ and ϒ(6S) χ φ, which are collected in the appendix. y considering the isospin symmetry, a general expression of the total amplitude of ϒ(6S) χ bj φ with J = 0,, is written as M Total ϒ(6S) χ bj φ = M (J j). (6) j Then the partial decay width reads Ɣ ϒ(6S) χbj φ = p φ 3 8π m M Total ϒ(6S) χ bj φ, (7) ϒ(6S) where the overline indicates the sum over polarizations of ϒ(6S), φ, and χ b (or χ ) and the factor 3 denotes the average over the polarization of initial ϒ(6S). In the case of ϒ(6S) χ bj ω, the expression of the partial decay width is given by Ɣ ϒ(6S) χbj ω = 3 8π p ω m ϒ(6S) A Total ϒ(6S) χ bj ω, (8) with a general expression of the total amplitude of ϒ(6S) χ bj ω as A Total ϒ(6S) χ bj ω = 4 A (J j), (9) j by considering the isospin and charge symmetry. The detailed expressions of A (J j) are collected in the appendix. 3 Numerical results gα δ + k δk α /m s k m s g βζ + k β kζ /m s k m s With the formulas listed in Sect. and the appendix, we estimate the hadronic loop contributions to the ϒ(6S) χ bj φ 3

5 Eur. Phys. J. C (07) 77 :65 Page 5 of 0 65 Table The coupling constants of ϒ(6S) interacting with () () Final state Decay width (MeV) Coupling constant GeV s s s s GeV s s Here, we also list the corresponding partial decay widths provided in Ref. [ together with ϒ(6S) χ bj ω(j = 0,, ) transitions. esides the masses taken from the Particle Data ook [, all the other input parameters we need are the coupling constants. Since the ϒ(6S) is above the threshold of () the coupling constants between ϒ(6S) and () () evaluated by the partial decay widths of ϒ(6S) () (), can be (). In Table we list the relevant partial decay widths given in Ref. [8 as well as the corresponding extracted coupling constants. The coupling constants relevant to the interactions between χ bj and () () in the heavy quark limit are related to one gauge coupling g given in Eq. (), i.e., g χb0 = 3g mχb0 m, g χb0 = 3 g mχb0 m, Fig. 7 The α dependence of the branching ratios of ϒ(6S) χ bj φ g χb = g mχb m m, mχ g χ = g mχb0 m, Fig. 8 The α dependence of the branching ratios of ϒ(6S) χ bj ω g χ = g m 3, g χ m = 4g mχ m, mχb0 where g = 3 f χb0 [8 and f χb0 = 75 ± 55 MeV is the decay constant of χ b0 [9. () Similarly, the coupling constants between φ or ω and can be extracted from Eq. (5), () g s s φ = g s s φ = βg V, f s s φ = f s s φ m s = λg V, g ω = g ω = βg V, f ω = f ω m = λg V, with β = 0.9 and λ = 0.56 GeV. Additionally, we have g V = m ρ /f π along with the pion decay constant f π = 3 MeV [ 4. With the above preparation, we can evaluate the branching ratiosoftheϒ(6s) χ bj φ and ϒ(6S) χ bj ω transi- tions. However, in our model, there still exists a free parameter α, which is introduced to parameterize the cutoff. Since the cutoff should not be too far from the physical mass of the exchanged mesons [7, in this work we set the range 0.65 α.5 for ϒ(6S) χ bj φ transitions and set 0.45 α.5 for ϒ(6S) χ bj ω transitions to present the numerical results. In Figs. 7 and 8, we illustrate the α dependence of the branching ratios of ϒ(6S) χ bj φ and ϒ(6S) χ bj ω, respectively, and in Figs. 9 and 0 we present the α dependence of the relative magnitudes among the branching widths of ϒ(6S) χ bj φ and ϒ(6S) χ bj ω, respectively. Varying α between 0.65 and.5 in ϒ(6S) χ bj φ, we have from Fig. 7, [ϒ(6S) χ b0 φ=( ) 0 6, [ϒ(6S) χ b φ=( ) 0 6, [ϒ(6S) χ φ=(. 5.8) 0 6, 3

6 65 Page 6 of 0 Eur. Phys. J. C (07) 77 : Fig. 9 The α dependence of the ratios R φ 0 = [ϒ(6S) χ b φ/[ϒ(6s) χ b0 φ, R φ 0 = [ϒ(6S) χ φ/[ϒ(6s) χ b0 φ and R φ = [ϒ(6S) χ φ/[ϒ(6s) χ b φ R0 ω = [ϒ(6S) χ ω [ϒ(6S) χ b0 ω 7.06, R ω = [ϒ(6S) χ ω [ϒ(6S) χ b ω.7. As shown in the numerical results on the ϒ(6S) χ bj φ decays, the partial decay widths of ϒ(6S) χ b0 φ and ϒ(6S) χ b φ are of the same order of magnitude, while the partial decay width of ϒ(6S) χ φ is one order of magnitude larger than those of ϒ(6S) χ b0 φ and ϒ(6S) χ b φ. On the other hand for the ϒ(6S) χ bj ω decays, the partial decay widths of ϒ(6S) χ b ω and ϒ(6S) χ ω are nearly of the same order of magnitude, while the partial decay width of ϒ(6S) χ b0 φ is one order of magnitude smaller than those of ϒ(6S) χ b ω and ϒ(6S) χ ω Fig. 0 The α dependence of the ratios R ω 0 = [ϒ(6S) χ b ω/[ϒ(6s) χ b0 ω, R ω 0 = [ϒ(6S) χ ω/[ϒ(6s) χ b0 ω and R ω = [ϒ(6S) χ ω/[ϒ(6s) χ b ω and for α varying from 0.45 to.5 in ϒ(6S) χ bj ω,we have from Fig. 8 [ϒ(6S) χ b0 ω=(0.5.8) 0 3, [ϒ(6S) χ b ω=( ) 0 3, [ϒ(6S) χ ω=( ) 0 3. In addition, some typical values for the relative magnitudes of the predicted branching ratios are obtained from Figs. 9 and 0, which are weakly dependent on the free parameter α, i.e., R φ 0 = [ϒ(6S) χ bφ [ϒ(6S) χ b0 φ 0.74, R φ 0 = [ϒ(6S) χ φ [ϒ(6S) χ b0 φ 3.8, R φ = [ϒ(6S) χ φ [ϒ(6S) χ b φ 4.43, R0 ω = [ϒ(6S) χ bω [ϒ(6S) χ b0 ω 4., 4 Summary In the past years, the anomalous hadronic transitions like ϒ(5S) ϒ(nS)π + π (n =,, 3) [ and ϒ(5S) χ bj ω (J = 0,, ) [ were reported by elle, which has stimulated theorists interest in revealing the underlying mechanism behind these phenomena [4 0. As a popular and accepted opinion, has been applied to explain why there exist anomalous transitions for ϒ(5S) [4 0. In addition, more predictions relevant to the ϒ(5S) transition were given in Refs. [,3. The main reason to introduce is that ϒ(5S) is the second observed bottomonium above the threshold, where the coupled-channel effect may become important, which was tested by the studies in Refs. [4 0,,3. It is obvious that this is not the end of the story. If is a universal mechanism existing in higher bottomonium transitions, we have reason to believe that this mechanism also plays an important role in higher bottomonium transitions. Considering the similarity between ϒ(6S) and ϒ(5S), where ϒ(6S) is the third bottomomium with open-bottom channels, we have focused on ϒ(6S) χ bj φ and ϒ(6S) χ bj ω hadronic decays. Using, we have estimated the branching ratios of ϒ(6S) χ bj φ and ϒ(6S) χ bj ω, which can reach up to 0 6 and 0 3, respectively. In the near future, elleii will be running near the energy range of ϒ(6S), which makes elleii have a great opportunity to find the χ bj φ and χ bj ω decay modes of ϒ(6S). If these rare decays are observed, the hadronic loop effects can be further tested. In this work, we have especially obtained the six almost stable ratios R φ 0, Rφ 0 and Rφ in addition to Rω 0, Rω 0 and Rω reflecting the relative magnitudes of the ϒ(6S) χ bj φ and 3

7 Eur. Phys. J. C (07) 77 :65 Page 7 of 0 65 ϒ(6S) χ bj ω decays, which are weakly dependent on our model parameter α. Thus, these obtained ratios are important observable quantities. We have suggested their experimental measurement, which is also a crucial test of our model. We notice the recent discussions of the status of SuperKEK and the future plan of taking data at the elleii experiment [30. Since the collision data on ϒ(6S) will be taken, we need to explore the possible interesting research issues about ϒ(6S). Our present work is only one step in a long march. Acknowledgements This project is supported by the National Natural Science Foundation of China under Grants Nos. 547, 75073, 37540, and , and by Chinese Academy of Sciences under the funding Y0460YQ0 and the agreement No. 05-H-0. XL is also supported by the National Program for Support of Young Top-notch Professionals. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP 3. Appendix As for the ϒ(6S) χ b φ transition, the amplitudes corresponding to Fig. are M ( ) = (π) 4 [ ig ϒ s s ɛ μ ϒ ((ik ) μ (ik ) μ ) [ f s s φ ε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) [ig s s χ b ɛζ χ b k m s k m s gσ ζ + q ζ q σ /m s q m F (q ), (0) s M ( ) = (π) 4 [ g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ɛφλ (ig s s φ g δσ (( ik ) λ (iq) λ ) + 4if s s φ (ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ig s s χ b ɛζ χ b g αδ + k α k δ /m s k m s g σζ + q σ q ζ /m s k m s q m F (q ), () s M ( 3) = (π) 4 [g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ ig s s φɛ φλ (( ik ) λ (iq) λ ) [ ig s s χ b ɛ ζ χ b g αζ + k α k ζ /m s k m s k m s q m s F (q ), () M ( 4) = (π) 4 [ig ϒ s s ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ f s s φε λρδσ ɛ λ φ (ip ) ρ (( ik ) δ (iq) δ ) [ ig s s χ b ɛ ζ χ b gασ + k α kσ /m s k m s gβ ζ + k ζ k β /m s k m s q m s F (q ). (3) As for the ϒ(6S) χ φ transition, the amplitudes corresponding to Fig. 3 are M ( ) = (π) 4 [ ig ϒ s s ɛ μ ϒ ((ik ) μ (ik ) μ ) [ ig s s φɛφλ (( ik ) λ (iq) λ ) [ g s s χ ɛχ ζη ( iq) ζ ( ik ) η k m s k m s q m F (q ), (4) s M ( ) = (π) 4 [ ig ϒ s s ɛ μ ϒ ((ik ) μ (ik ) μ ) [ f s s φ ε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) [ ig s s χ ε ζωκξɛχ ζη (ip 3 ) κ ( iq) η ( ik ) ξ k m s k m s gσω + q σ q ω /m s q m F (q ), (5) s M ( 3) = (π) 4 [ g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f s s φε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) [ g s s χ ɛχ ζη ( iq) ζ ( ik ) η gσ α + k αk σ /m s k m s k m s q m F (q ), (6) s 3

8 65 Page 8 of 0 Eur. Phys. J. C (07) 77 :65 M ( 4) = (π) 4 [ g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ɛφλ (ig s s φ g δσ (( ik ) λ (iq) λ ) + 4if s s φ (ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ ig s s χ ε ζωκξɛχ ζη (ip 3 ) κ ( iq) η ( ik ) ξ g αδ k α k δ /m s k m s k m s gω σ + q σ q ω /m s q m F (q ), (7) s M ( 5) = (π) 4 [g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ ig s s φɛφλ (( ik ) λ (iq) λ ) [ig s s χ ε ζωκξ ɛχ ζη (ip 3 ) κ ( iq) ξ ( ik ) η gω α + k αkω /m s k m s k m s q m F (q ), (8) s M ( 6) = (π) 4 [g ϒ s s εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f s s φ ε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) [g s s χ ɛζη χ (g ζκ g ηξ + g ηκ g ζξ ) gκ α + k αkκ /m s k m s k m s gσξ + q σ q ξ /m s q m F (q ), (9) s M ( 7) = (π) 4 [ig ϒ s s ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ f s s φε λρδσ ɛφ λ (ip ) ρ (( ik ) δ (iq) δ ) [ig s s χ ε ζωκξ ɛχ ζη (ip 3 ) κ ( iq) ξ ( ik ) η gασ + k αkσ /m s g βω + k β kω /m s k m s k m s q m F (q ), (30) s M ( 8) = (π) 4 [ig ϒ s s ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ɛφλ (ig s s φ g δσ (( ik ) λ (iq) λ ) + 4if s s φ (ip ) ρ (g λδ g ρσ g λσ g ρδ )) [g s s χ ɛζη χ (g ζκ g ηξ + g ηκ g ζξ ) gα δ + k δk α /m s k m s g βκ + k β kκ /m s k m s gξ σ + q σ q ξ /m s q m F (q ). (3) s As for the ϒ(6S) χ b0 ω transition, the amplitudes corresponding to Fig. 5 are A (0 ) = (π) 4 [ ig ϒ ɛ μ ϒ ((ik ) μ (ik ) μ ) [ ig ω ɛωλ (( ik ) λ (iq) λ )[ g χb0 k m k m q m F (q ), (3) A (0 ) = (π) 4 [ g ϒ ε μναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) [ g χb0 gσ α + k αk σ /m k m k m q m F (q ), (33) A (0 3) = (π) 4 [g ϒ εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) gα ζ + k α k ζ [ g χ b0 /m k m k m gσ ζ + qσ q ζ /m q m F (q ), (34) A (0 4) = (π) 4 [ig ϒ ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ɛωλ (ig ωg δσ (( ik ) λ (iq) λ ) + 4if ω(ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ g χ b0 gα δ + k δk α /m k m g βζ β + k kζ /m k m g σζ + q σ q ζ /m q m F (q ). (35) 3

9 Eur. Phys. J. C (07) 77 :65 Page 9 of 0 65 As for the ϒ(6S) χ b ω transition, the amplitudes corresponding to Fig. 5 are A ( ) = (π) 4 [ ig ϒ ɛ μ ϒ ((ik ) μ (ik ) μ ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) [ig χ b ɛχ ζ b k m k m gσ ζ + q ζ q σ /m q m F (q ), (36) A ( ) = (π) 4 [ g ϒ ε μναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ɛωλ (ig ωg δσ (( ik ) λ (iq) λ ) + 4if ω(ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ig χ b ɛ ζ χ b g αδ + k α k δ /m k m k m A ( 3) = g σζ + q σ q ζ /m q m F (q ), (37) (π) 4 [g ϒ εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ ig ω ɛωλ (( ik ) λ (iq) λ ) [ ig χ b ɛχ ζ b k m g αζ + k α k ζ /m k m q m F (q ), (38) A ( 4) = (π) 4 [ig ϒ ɛμ ϒ (g ναg μβ (ik ) ν g μα g νβ (ik ) ν g αβ ((ik ) μ (ik ) μ )) [ f ωε λρδσ ɛ λ ω (ip ) ρ (( ik ) δ (iq) δ ) [ ig χ b ɛ ζ χ b gασ + k α kσ /m k m gβ ζ + k ζ k β /m k m q m F (q ). (39) As for the ϒ(6S) χ ω transition, the amplitudes corresponding to Fig. 6 are A ( ) = (π) 4 [ ig ϒ ɛ μ ϒ ((ik ) μ (ik ) μ ) [ ig ω ɛωλ (( ik ) λ (iq) λ ) [ g χ ɛχ ζη ( iq) ζ ( ik ) η k m k m q m F (q ), (40) A ( ) = (π) 4 [ ig ϒ ɛ μ ϒ ((ik ) μ (ik ) μ ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) [ ig χ ε ζωκξ ɛχ ζη (ip 3 ) κ ( iq) η ( ik ) ξ k m k m gσω + q σ q ω /m q m F (q ), (4) A ( 3) = (π) 4 [ g ϒ ε μναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) [ g χ ɛχ ζη ( iq) ζ ( ik ) η gσ α + k αk σ /m k m k m q m F (q ), (4) A ( 4) = (π) 4 [ g ϒ ε μναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ɛωλ (ig ωg δσ (( ik ) λ (iq) λ ) + 4if ω(ip ) ρ (g λδ g ρσ g λσ g ρδ )) [ ig χ ε ζωκξ ɛχ ζη (ip 3 ) κ ( iq) η ( ik ) ξ g αδ k α k δ /m k m k m gω σ + q σ q ω /m q m F (q ), (43) A ( 5) = (π) 4 [g ϒ εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ ig ω ɛωλ (( ik ) λ (iq) λ ) [ig χ ε ζωκξ ɛχ ζη (ip 3 ) κ ( iq) ξ ( ik ) η gα ω + k αk ω/m k m k m q m F (q ), (44) A ( 6) = (π) 4 [g ϒ εμναβ ɛ ϒμ ( ip ) ν ((ik ) β (ik ) β ) [ f ωε λρδσ ɛω λ (ip ) ρ (( ik ) δ (iq) δ ) [g χ ɛχ ζη (g ζκ g ηξ + g ηκ g ζξ ) gα κ + k αk κ/m k m k m gσξ + q σ q ξ /m q m F (q ), (45) 3

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