The QCD anomaly coupling of η′-g-g is treated as the axial vector current triangle anomaly. By assuming the divergent axial vector coupling of the η′ meson with the quark line in the triangle diagram, we calculate the QCD anomaly of η′-g-g as well as the QED anomaly of η′-γ-γ with only one common parameter Kη′. We obtain consistent results of the branching ratios for ..

Muta, Taizo

Yang, Mao-Zhi

Hiroshima University Institutional Repository

arXiv:hep-ph/9902275 v1   9 Feb 1999QCD Anomaly Coupling for the η′ − g − g Vertexin Inclusive Decay B → η′XsTaizo Muta and Mao-Zhi YangDepartment of Physics, Hiroshima University, Higashi-Hiroshima,Hiroshima 739-8526, Japan∗March 15, 2006PACS Numbers: 13.25.Hw, 11.30.Er, 13.40.Hq, 12.38.AwAbstractThe QCD anomaly coupling of η′ − g − g is treated as the axialvector current triangle anomaly. By assuming the divergent axialvector coupling of the η′ meson with the quark line in the trianglediagram, we calculate the QCD anomaly of η′−g−g as well as theQED anomaly of η′ − γ − γ with only one common parameter κη′ .We obtain consistent results of the branching ratios for B → η′Xs,J/ψ → η′γ and η′ → γγ when comparing with experimental data.∗mailing address1The first observation of B → η′Xs decays [1, 2] has stimulated much theoreticalinterests. A number of interpretations have been proposed to explain the observedlarge decay branching fraction. One of the interpretations is based on b → cc¯sdecay with the assumption of the intrinsic cc¯ content in the η′ wave function[3],while the others are based on the b→ sg∗ penguin transition followed by g∗ → η′gthrough QCD anomaly [4, 5, 6, 7, 8, 9]. The observed recoil mass spectrum favorsthe second interpretation[2]. Thus the QCD anomaly coupling of the η′ meson withgluon fields is important in the B → η′Xs decay process. However there are still noconsensus among the authors on the second interpretation. Some of them concludethat the calculation within the standard model (SM) is sufficient to account for theexperimental data[4, 7, 9] while the others[5, 6] conclude not, stressing that newphysics is needed for the proper interpretation. The key point of the confusingsituation comes from the QCD anomaly itself. Refs.[4, 5, 7] stress that the QCDanomaly coupling for the η′ − g − g vertex is highly nonperturbative so that it isunpredictable. Because the η′− g− g vertex is treated differently in Refs.[4, 5, 6, 7],the different results of the branching ratio of B → η′Xs are drawn.In the present paper, we try to make clear the above confusing situation onthe η′ − g − g coupling. Given the unique properties of the axial current triangleanomaly which has first been discovered in QED and successfully solved the problemof the pi0 → γγ decay[10] the η′ − g − g coupling is also treated as the axial currentanomaly here (Fig.1). The vertex of the η′ meson coupling with the quark line inthe triangle diagram is assumed to be given by iκη′/pγ5 which comes from the PCAChypothesis[11]. Here κη′ is introduced as a coupling parameter. It is resonable totreat κη′ as a constant, because it only involves the inner properties of the η′ meson.2xHHHHHHY?*k1 -k2 -q+k1qq+pp -igTaγµigTbγνi/pγ5κη′xHHHHHHY?*k2 -k1 -q+k2qq+pp -igTbγνigTaγµi/pγ5κη′Figure 1: The lowest-order diagrams which contribute to QCD anomaly coupling ofη′ − g − g. Higher order diagrams do not contribute to the anlmaly term accordingto the Adler-Bardeen theorem[12]The routing of the loop momentum q is shown in Fig.1. We write the amplitudeT µνλ directly by following the Feynman rule,T µνλ =∫d4q(2pi)4(−1){Tr[i/q −miγλγ5κη′i/q + /p−migγνTbi/q + /k1 −migTaγµ]+k1 ↔ k2µ ↔ ν(1)Maintaining the vector current Ward indentities k1µTµνλ = 0 and k2νTµνλ = 0 weobtain,pλTµνλ = 2mT µν − 12pi2(12δabg2κη′)εµνρσk1ρk2σ, (2)whereT µν =(12δabg2κη′) ∫d4q(2pi)4(−1){Tr[1/q −mγ51/q + /p−mγν 1/q + /k1 −mγµ]+k1 ↔ k2µ ↔ ν(3)3For the details of the derivation of eq.(2) and (3), the readers may refer to Ref.[13].After a few steps of algebraic calculation eq.(3) can be finally converted into thefollowing Feynman integration.T µν =(12δabg2κη′)8mI(k21, k22, p2)εµνρσk1ρk2σ, (4)whereI(k21, k22, p2) =−116pi2∫ 10dx∫ 1−x0dy1(x2 − x)k21 + (y2 − y)k22 − 2xyk1 · k2 +m2. (5)Now some properties of eq.(2) should be pointed out. The second term in theright hand side of eq.(2) violates the axial current Ward identity pλTµνλ = 2mT µν ,it is called anomaly term. The factor 12pi2(12δabg2κη′)is momentum independent.Substituting eq.(4) into eq.(2) we getpλTµνλ = [16m2I(k21, k22, p2)− 12pi2](12δabg2κη′)εµνρσk1ρk2σ. (6)After practising the numerical calculation through the use of the computer wefind for the whole range of the momentum squared k21 and k22, which is favored bythe decay B → η′Xs, the first term in the square brackets of eq.(6) is almost twoorders of magnitude smaller than the second one. Hence we can safely drop the firstterm. Now the QCD anomaly coupling for the η′ − g − g vertex is finally given byAµν = −αs(µ)κη′piDδabεµνρσk1ρk2σ, (7)where D =√3cosθ which is introduced by taking into account the contributions ofthree quarks contained in η′ meson. The angle θ is the pseudoscalar mixing angledefined by η′ = η0cosθ + η8sinθ and our choice of the angle is θ = −19.50.4We use eq.(7) to calculate the branching ratio for B → η′Xs. We use thebranching ratio of b → sgη′ to estimate the inclusive process. The strong penguininduced b→ s current is [14]Gf√2gs4pi2Vts¯Ta{∆F1(q2γµ − qµ)L− F2iσµνqνmbR}b, (8)where ∆F1 ≃ −5, F2 ≃ 0.286 and Vt = v∗tsvtb. For the convenience of expressing theresult for the branching ratio of b→ sgη′ we first define some variables: x ≡ m2x/m2bwith mx the invariant mass of the hadronic system in recoil from the η′ meson inthe final state, y ≡ k21/m2b with k1 the momentum of the virtual gluon g∗ whichconnects the penguin diagram with the η′ meson, x′ ≡ m2η′/m2b and x′s ≡ m2s/m2b .The branching ratio can be expressed asBr(b→ sgη′) = 1ΓBG2f |Vt|2m5b192pi3g2s(mb)16pi4m2b4(αs(µ)κη′piD)2×∫dx∫dy{12|∆F1|2[−2(x− x′x)2y + (1− y − x′s)(y − x′)(2x+ y − x′ − 2x′s)]−Re(∆F1F ∗2 )(1− y − x′s)(y − x′)2/y+12|F2|2[2(x− x′s)2y2 − (1− y − x′s)(y − x′)(2(x− x′s)y − (y − x′)(1− x′s))]/y2}. (9)It should be stressed that our result of the anomaly coupling for the η′ − g − gvertex is obtained by assuming the divergent axial vector coupling between η′ andthe quark line in the triangle diagram. If we only use it to explain one experiment,the validity of our method cannot be tested. So we should use the same method andthe same parameter κη′ to other processes. Now we will use it to calculate η′ → γγdecay which is an electromagnetic decay process, so it is completely different from5η′ − g − g coupling in which strong interaction is involved. Calculating the similartriangle diagrams as shown in Fig.1, where the coupling iκη′/pγ5 is the same, onlythe strong coupling igTaγµ, igTbγν are changed to be the QED coupling iQeγµ andiQeγν . We find the amplitude of η′ → γγ isAγγ = −2ακη′piD′εµνρσk1ρk2σε∗µε∗ν . (10)where α is the fine-structure constant, i.e., α = 1/137,D′ =[1√3(Q2u +Q2d +Q2s)cosθ +1√6(Q2u +Q2d − 2Q2s)sinθ]·Nc ≃ 73√6,where Nc = 3 is the color number. Finally we getΓη′→γγ =164pi(143√6ακη′pi)2m3η′ . (11)Comparing eq.(11) with experimental data, Γexp.η′→γγ = (4.28± 0.43)× 10−6GeV [15],we findκη′ = 7.06± 0.35GeV −1. (12)Substitute the value of κη′ into eq.(9), and take mb = 4.8GeV, ms = 0.15GeV,1ΓBG2f|Vt|2m5b192pi3≈ 0.2, we can get,Br(b→ sgη′) = (4.9± 0.5)× 10−4, (13)here the error bar ±0.5 comes from ±0.35 in eq.(12), and the experimental cut hasbeen taken into account in eq.(13). Comparing eq.(13) with the experimental dataBr(B → η′Xs) = (6.2± 2.0)× 10−4 [2] we find that eq.(13) is fairly consistent withthe data.Some remarks should be given. First, to get eq.(13), we have used the runningstrong coupling constant αs(µ) in eq.(7)[16]. Althought it is samller than the re-sult which is obtained without taking into account the running of αs(µ), it is well6within the experiment error bar. In contrast, it will be too large to account for theexperimental data if we do not take into account the running of αs(µ). Second, atµ = mη′ we find the anomaly coupling in eq.(7) isαs(mη′ )κη′piD.= 1.8GeV −1. It is alsoconsistent with the anomaly coupling for the η′−g−g vertex which AS[4] extractedfrom the experimental data of Br(J/ψ → η′γ). Hence the QCD anomaly couplingfor the η′ − g − g vertex derived by calculating the triangle diagram can also givethe correct result of Br(J/ψ → η′γ). Third, in general, it is believed that the QCDanomaly coupling of η′ − g − g is nonperturbative and hence it is unpredictable.But now our calculation shows that the nonperturbative part can be successfullyseparated, which is obsorbed into the coupling parameter kη′ . With the parameterkη′ the strong coupling of η′−g−g can be predicted from the knowledge of η′ → γγ,which is an electromagnetic decay process. It is very interesting to note that theQCD anomaly and QED anomaly can be treated in an uniform way.In summary, by calculating AVV (Axial vector-Vector-Vector current) trianglediagram, we find that QCD anomaly coupling for the η′ − g − g vertex and QEDanomaly coupling η′−γ−γ can be treated with one parameter κη′ . The calculationof these three decay process B → η′Xs, J/ψ → η′γ, and η′ → γγ can be consistentwith experiment at the same time. SM is sufficient to account for B → η′Xs datawithin the present experimental error bars.The outhors would like to thank Dr. C. D. Lu¨ and Dr. T. Morozumi for usefuldiscussions. One of us (MZY) thanks Japan Society for the Promotion of Science(JSPS) for financial support.References7[1] CLEO Collaboration, B. Behrens, talk presented at the conference on B physicsand CP violation, Honolulu, Hawaii, 1997; CLEO Collaboration, F. Wuerthwin,talk presented at Rencontre de Moriond, QCD and High Energy Hadronic Inter-actions, Les Arcs, 1997; D.H. Miller, talk presented at EPS-HEP97, Jerusalem,Israel, 1997; I. Smith, talk presented at the 1997 Aspen Winter Physics Con-ference on Particle Physics, 1997;[2] CLEO Collaboration, T.E. Browder et al., Phys. Rev. Lett. 81, 1786 (1998).[3] I. Halperin and A. Zhitnitsky, Phys. Rev. D56, 7247 (1997); I. Halperin andA. Zhitnitsky, Phys. Rev. Lett. 80, 438 (1998); F. Yuan and K-T. Chao, Phys.Rev. D56, 2495 (1997).[4] D. Atwood and A. Soni, Phys. Lett. B405, 150 (1997); D. Atwood and A. Soni,Phys. Rev. Lett. 79, 5206 (1997).[5] W-S. How and B. Tseng, Phys. Rev. Lett. 80, 433 (1998).[6] A. L. Kagan and A. A. Petrov, hep-ph/9707354.[7] H. Fritzsch, Phys. Lett. B415, 83 (1997).[8] D-S. Du and M-Z. Yang, Phys. Rev. D57, 5332 (1998).[9] X-G. He and G-L. Lin, hep-ph/9809204; X-G. He, hep-ph/9812247;[10] S.L. Adler, Phys. Rev. 177, 2426 (1969); J.S. Bell and R. Jackiw, Nuovo Ci-mento A60, 47 (1969).8[11] M. Gell-mann and M. Levy, Nuovo Cimento 16, 705 (1960); J. Berstein, S.Fubini, M. Gell-mann and W. Thirring, Nuovo Cimento 17, 757 (1960); ChouKuang-chao, Soviet Physics, 12, 492 (1961) [JETP 39, 703 (1963)].[12] A.L. Adler and W.A. Bardeen, Phys. Rev. 182, 1517 (1969).[13] Ta-Pei Cheng and Ling-Fong Li, Gauge theory of elementary particle physics,Clarendon Press, Oxford, 1984.[14] W-S. How and A. Soni, Phys. Rev. Lett. 59, 1521 (1987); W-S. How, Nucl.Phys. B308, 561 (1988).[15] C. Caso et al., Particle Data Group, The European Physical Journal C3, 1(1998).[16] The running of the effective QCD coupling constant isα(f)s (µ) =4piβ0ln(µ2/Λ2f)[1− β1β20lnln(µ2/Λ2f)ln(µ2/Λ2f)]with β0 = 11 − 23f and β1 = 102 − 383 f , where f is the number of the quarkswhich are active in the scale of µ. Λ4 = 300MeV is used in this paper, and tokeep the running of α(f)s (µ) continuous at µ = mc = 1.5GeV , Λ3 = 353MeV istaken here.9

QCD anomaly coupling for the η′-g*-g vertex in inclusive decay B→ η′Xs

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QCD anomaly coupling for the η′-g*-g vertex in inclusive decay B→ η′Xs

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