Lepton flavor violation decays of vector mesons in unparticle physics
Ke-Sheng Sun, Tai-Fu Feng, Li-Na Kou, Fei Sun, Tie-Jun Gao, Hai-Bin Zhang
aa r X i v : . [ h e p - ph ] D ec April 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
LEPTON FLAVOR VIOLATION DECAYS OF VECTOR MESONSIN UNPARTICLE PHYSICS
KE-SHENG SUN † , ‡ , ∗ , TAI-FU FENG ‡ , † , LI-NA KOU † , FEI SUN † ,TIE-JUN GAO ‡ , † , HAI-BIN ZHANG ‡ , †† Department of Physics, Dalian University of Technology, Dalian 116024, China ‡ Department of Physics, Hebei University, Baoding 071002,China ∗ [email protected] Received (Day Month Year)Revised (Day Month Year)We investigate the lepton flavor violation decays of vector mesons in the scenario of theunparticle physics by considering the constraint from µ − e conversion. In unparticlephysics, the predictions of LFV decays of vector mesons depend strongly on the scaledimension d U . The predictions of LFV decays of vector mesons can reach the detectivesensitivity in experiment in region of 3 ≤ d U ≤
4, while the prediction of µ − e conver-sion rate can meet the experimental upper limit. For the searching of the lepton flavorviolation processes of charged lepton sector in experiment, the process Υ → eµ may bea promising one to be observed. Keywords : Lepton flavor violating; Unparticle.PACS Nos.:13.20.-v, 12.60.-i
1. Introduction
During the last decades, searching for Lepton Flavor Violation (LFV) processes incharged lepton sector, as an evidence to discover new physics beyond the Stan-dard Model (SM), have attracted a great deal of attention. Although nonzero neu-trino masses supported by the neutrino oscillation experiments 1 , , , , , , , ,
12 These new sources are mainlyoriginated from the interactions between the SM particles and new particles be-yond SM. Instead, Georgi proposes an alternative scenario that there could be asector that is exactly scale invariant and very weakly interacting with the sectorsin SM.13 ,
14 There are no particles in such a sector in space-times spaces cause noparticles states with a nonzero mass exist. In general, the scale invariant sector orthe so-called unparticle has a scale dimension of fractional number rather than an pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. integral number. The interactions between the unparticle and the SM particles inlow energy effective theory can lead to various interesting features in LFV processesand other phenomenologies. In unparticle physics, the unparticle can interact withdifferent flavors of SM leptons and this indicates that the LFV processes can happenat tree level. There have been many studies of LFV processes in unparticle physics.Such as, µ → e ,15 , µ → eγ ,17 µ − e conversion,17 M → ll ′ ,18 e + e − → ll ′ ,18 J/ Ψ → ll ′ ,19 r → ll ′ ,20 , l → l ′ γγ ,20 , τ → l ( V , P ),22 etc..The study of LFV processes involving vector mesons is an effective way maybeto search for new physics beyond the SM, and the SND Collaboration at the BINP(Novosibirk) presents an upper limit on the φ → e + µ − branching fraction ofBR( φ → e + µ − ) ≤ × − .23 Additionally, using a sample of 5 . × J/ Ψevents collected with the BESII detector, Ref. 24 obtains the upper limits onBR( J/ Ψ → µτ ) < . × − and BR(Υ → µτ ) < . × − at the 90% con-fidence level (C.L.). Adopting the data collected with the CLEO III detector, theauthors of Ref. 25 estimate the upper limits on BR (Υ(1 S ) → µτ ) < . × − ,BR (Υ(2 S ) → µτ ) < . × − and BR (Υ(3 S ) → µτ ) < . × − respectivelyat the 95% C.L. In literatures, several stringent limits on LFV decays of vectormesons are derived in a model independent way. Assuming that a vector boson M i couples to µ ∓ e ± and e ∓ e ± , the authors of Ref. 26 deduce some upper bounds onthe LFV decays of vector mesons by a consideration of the experimental constrainton the process µ → e . Under a similar assumption that a vector meson M i couplesto µ ∓ e ± and nucleon-nucleon, Ref. 27 and Ref. 28 study the LFV decays of vectormesons by taking account of the experimental constraint on µ − e conversion.In this paper, we investigate the LFV decays of vector mesons in unparticlephysics by the consideration of constraint on µ − e conversion. In Section.2, wefirstly provide a brief introduction to the unparticle physics and corresponding in-teraction Lagrangian in effective field theory. Then we derive the analytic resultsof the amplitude in detail. The numerical analysis and discussion are presented inSection.3, and the conclusion is drawn in Section.4.
2. Formalism
In very high energy, as it is proposed by Georgi,13 ,
14 the theory is composed ofthe SM fields and the fields of a theory with a nontrivial IR fixed point, whichis called Banks-Zaks ( BZ ) fields.29 The two fields can interact by the exchangeof particles with a large mass scale M U ≫ T eV . Below the scale M U , there arenonrenormalizable couplings involving both standard model fields and Banks-Zaksfields suppressed by powers of M U . The interaction between SM field and BZ fieldhas the form: 1 M d SM + d BZ − U O SM O BZ , (1)where O SM is an operator with a mass dimension d SM corresponding to SM fieldsand O BZ is an operator with a mass dimension d BZ corresponding to BZ fields. Inpril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Lepton flavor violation decays of vector mesons in unparticle physics effective field theory, below the scale Λ U , the BZ operators would match onto theunparticle operators, and Eq.(1) can be viewed as the interaction between SM fieldand unparticle field: C U Λ d BZ − d U U M d SM + d BZ − U O SM O U , (2)where C U is a coefficient function, d U denotes the scaling dimension of the unparticleoperator O U .For simplicity, it is convenient to define: λ = C U Λ d BZ U M d SM + d BZ − U . (3)Then, in effective theory, the couplings of the scalar and vector unparticles to SMfermions (leptons or quarks) are generally given by the following effective operators: λ SSff ′ Λ d U − U ¯ f f ′ O U , λ SPff ′ Λ d U − U ¯ f γ f ′ O U , λ SVff ′ Λ d U U ¯ f γ µ f ′ ∂ µ O U , λ SAff ′ Λ d U U ¯ f γ µ γ f ′ ∂ µ O U ,λ V Sff ′ Λ d U U ¯ f f ′ ∂ µ O µ U , λ V Pff ′ Λ d U U ¯ f γ f ′ ∂ µ O µ U , λ V Vff ′ Λ d U − U ¯ f γ µ f ′ O µ U , λ V Aff ′ Λ d U − U ¯ f γ µ γ f ′ O µ U , (4)where λ S,P,V,Aff ′ are dimensionless coefficients. S, P, V and A stand for scalar field,pseudo-scalar field, vector field and axial vector field, respectively. f and f ′ denoteSM fermions, O U and O µ U denote scalar and vector unparticle fields. The propagatorof scalar unparticle field has the form14 , Z e iP · x d x h | T [ O U ( x ) O U (0) | i = i A d U d U π ) 1( − P − iǫ ) d U − (5)If the vector unparticle field is assumed to be transverse, the propagator can beenwritten as: Z e iP · x d x h | T [ O µ U ( x ) O ν U (0) | i = i A d U d U π ) − g µν + P µ P ν ( − P − iǫ ) − d U (6)where A d U is defined by: A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) (7)For vector mesons, only the vector current ¯ f γ µ f ′ couples to vector mesons. Thetree level Feynman diagram is presented in Fig.1. The amplitude for Fig.1 can bewritten as: M Q = λ V Vbb λ V Veµ Λ d U − U A d U d U π ) ¯ υ ( p ) γ µ u ( p ) g µν − p µ p ν p − d U ) ¯ u ( p ) γ ν υ ( p ) . (8)In the quark picture, mesons are composed of a quark and an antiquark. We adopt aphenomenological model where the amplitude of hard process involving a s-wave me-son can be described by the matrix elements of gauge-invariant nonlocal operators,pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. feynman diagram b µ ¯ b e Fig. 1. The tree level diagram for LFV process Υ → eµ , the double dashed line denotes the vectorunparticle field. which are sandwiched between the vacuum and the meson states. The distributionamplitude of vector meson Υ in leading-order is defined through the correlationfunction31 , , h | ¯ b i α ( y ) b j β ( x ) | Υ( p ) i = δ ij N c Z du e − i [ upy +(1 − u ) px ] h f Υ m Υ /εφ k ( u )+ i σ µ ′ ν ′ f T Υ ( ε µ ′ p ν ′ − ε ν ′ p µ ′ ) φ ⊥ ( u ) i βα , (9)where Nc is the number of colors, ε is the polarization vector, f Υ and f T Υ are thedecay constants, φ k and φ ⊥ are the leading-twist distribution functions correspond-ing to the longitudinally and transversely polarized meson, respectively. Since theleading-twist light-cone distribution amplitudes of meson are close to their asymp-totic form,34 so we set φ k = φ ⊥ = φ ( u ) = 6 u (1 − u ).Then, at hadron level, using Eq.(9), the amplitude is rewritten as M H = λ V Vbb λ V Veµ Λ d U − U A d U m Υ f Υ N c sin( d U π ) ε ν p − d U ) ¯ u ( p ) γ ν υ ( p ) . (10)In the frame of center of mass, using the summation formula X λ = ± , ε µλ ( p ) ε ∗ νλ ( p ) ≡ − g µν + p µ p ν m , (11)we get |M H | = | λ V Vbb λ V Veµ Λ d U − U A d U m Υ f Υ N c sin( d U π ) | m − m e − m µ ) − m e m µ m − d U )Υ . (12)Finally,we express the branching ratio of process Υ → eµ as Br (Υ → eµ ) = p [ m − ( m e + m µ ) ][ m − ( m e − m µ ) ]16 πm Γ Υ × |M H | , (13)where Γ Υ is the total decay width. The branching ratios for other LFV processes ofvector mesons can be formulated in a similar way.pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Lepton flavor violation decays of vector mesons in unparticle physics
3. Numerical Analysis
Taking account of the constraint on the LFV processes µ → eγ and µ − e conversionin nuclei, we will study the LFV decay of Υ → eµ in unparticle physics firstly.Under the assumption of the unparticle couplings with the SM fermions in Eq.(4)are universal: λ KKff ′ = ( λ k ,f = f ′ κλ k ,f = f ′ (14)where κ > µ → eγ and µ − e conversion in various nuclei in region of 1 < d U <
2, 1TeV < Λ U < d U deduced from experimentalbound on µ − e conversion is more stringent. Therefore, we will study the LFVdecays of vector mesons by a consideration of µ − e conversion in unparticle physics.The formula for the µ − e conversion rate with the pure vector coupling betweenSM fermions and unparticle is given by: CR ( µ − e, N ucleus ) = m µ α Z eff F p π Z [ λ V Veµ λ V Vqq A d U d U π ) 1Λ U ( m µ Λ U ) d U − ] × | Z X q G ( q,p ) V + N X q G ( q,n ) V | capt , (15)where Z and N denote the proton and neutron numbers in a nucleus, F p is thenuclear form factor and Z eff is an effective atomic charge, G ( q,p ) V and G ( q,n ) V arenuclear matrix elements relevant to proton and neutron.Here, as in Ref. 17, taking Λ U = 10TeV, λ V Vbb = 0 .
001 and λ V Veµ = 0 . → eµ ) versus d U and CR( µ − e, T i ) versus d U in region of 1 ≤ d U ≤ µ − e, T i ), the dot linedenotes the prediction of BR(Υ → eµ ). The horizontal lines correspond to 10 − and4 . × − , which are the experimental sensitivity of LFV decays of vector mesonsand the experimental bound on µ − e conversion rate respectively. The followingnumerical values are used36 , , m Υ = 9 . GeV, f Υ = 715 M eV, Γ Υ = 54 KeV,F p = 0 . , Z eff = 17 . , Γ capt = 1 . × − . (16)As we can see from Fig.2, the predictions of both BR(Υ → eµ ) and CR( µ − e, T i ) depend strongly on the scaling dimension d U . The value of dimension d U isconstrained to near 2 or more larger and the relevant prediction of BR(Υ → eµ ) ishighly suppressed to reach the experimental sensitivity.There is an interesting feature in Fig.2 that the prediction of BR(Υ → eµ )is less than CR( µ − e, T i ) in region of 1 ≤ d U ≤
3, however, in region of 3 ≤ d U ≤
4, the prediction of BR(Υ → eµ ) is larger than CR( µ − e, T i ). Consideringpril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. -70 -60 -50 -40 -30 -20 -10 B R o r CR d u Fig. 2. The BR(Υ → eµ ) and CR( µ − e, T i ) vary as a function of d U , where the solid line denotesthe prediction of CR( µ − e, T i ), the dot line denotes the prediction of BR(Υ → eµ ), the horizontallines correspond to 10 − and 4 . × − , which are the experimental sensitivity of LFV decaysof vector mesons and the experimental bound on µ − e conversion rate respectively. Λ U = 10TeV, λ V Vbb = 0 .
001 and λ V Veµ = 0 .
003 are assumed. the experimental bound on CR( µ − e, T i ) is O (10 − ), it is impossible to makethe prediction of BR(Υ → eµ ) reach the experimental sensitivity by resetting thecouplings | λ V Vbb λ V Veµ | in region of 1 ≤ d U ≤
3. Nevertheless, in region of 3 ≤ d U ≤ | λ V Vbb λ V Veµ | , it is available to get that not only theprediction of CR( µ − e, T i ) is compatible with the experimental bound, but also theprediction of BR(Υ → eµ ) is large enough to be detected in experiment at presentor in near future. Therefore, we will investigate the LFV decays of vector mesonsin region of 3 ≤ d U ≤
4. In addition, the constraint of d U ≥ U µ have d V ≥
3, with d V = 3 if and only if theoperator is a conserved current, ∂ µ U µ = 0.For the aim of enhancing the prediction of BR(Υ → eµ ) to be detectable inexperiment, the value of the couplings | λ V Vbb λ V Veµ | would be very large. However,we can also investigate the LFV decays of vector mesons in a way independent ofthe couplings | λ V Vbb λ V Veµ | . Considering the µ − e conversion in T i nucleus, let usdefine the fraction R ( X ) by: R ( X ) = BR ( X → eµ ) CR ( µ − e, T i ) , (17)pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Lepton flavor violation decays of vector mesons in unparticle physics where X would be any vector mesons: ρ , ω , φ , J/ Ψ or Υ. From Eq.(12), Eq.(13) andEq.(15), we can see that coefficient λ V Veµ and mass scale Λ U can be canceled out in R ( X ). As for the unparticle couplings with the quarks are universal, i.e., λ V Vbb = λ V Vss = λ V Vcc ≃ λ V Vuu + λ V Vdd √ ≃ λ V Vuu − λ V Vdd √ , (18)the couplings listed in Eq.(18) would also be canceled out in R ( X ) due to the samevalue setting in both numerator and denominator in Eq.(17). Therefore, R ( X ) wouldonly be a function of scaling dimension d U . Using Eq.(12), Eq.(13) and Eq.(15), therelation between R ( X ) and d U can be shown in a simple form: R ( X ) ∝ ( m X m µ ) d U − , (19)where m X m µ > R ( X ) and d U in Eq.(19) is still reliable. -14 -12 -10 -8 -6 -4 -2 R ( X ) d U Fig. 3. The fraction R ( X ) varies as a function of d U , where, from the bottom up, the lines standfor R ( ρ ), R ( ω ), R ( φ ), R ( J/ Ψ) and R (Υ), respectively. In Fig.3, we display the fraction R ( X ) varies as a function of d U in region of3 ≤ d U ≤
4, where, from the bottom up, the lines stand for R ( ρ ), R ( ω ), R ( φ ), R ( J/ Ψ)and R (Υ), respectively. The parameters relevant to different mesons are listedpril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. below36 , , m ρ = 775 M eV, f ρ = 209 M eV, Γ ρ = 149 M eV,m ω = 782 M eV, f ω = 195 M eV, Γ ω = 8 . M eV,m φ = 1 . GeV, f φ = 231 M eV, Γ φ = 4 . M eV,m J/ Ψ = 3 . GeV, f J/ Ψ = 405 M eV, Γ J/ Ψ = 92 . KeV.
It displays in Fig.3 that R ( X ) increases as d U grows. For light flavor mesons, thefraction R ( X ) is very small. However, for heavy flavor mesons, the fraction R ( X )is large and we can get a large BR( X → eµ ) to be detectable in experiment andcompatible with the constraint on µ − e conversion.From Eq.(17) we can express the branching ratio of X → eµ as: BR ( X → eµ ) = R ( X ) × CR ( µ − e, T i ) . (20)Using Eq.(20), we give the predictions on branching ratios of LFV decays of vectormesons in Tab.1 with d U = 3, d U = 3 . d U = 4, where CR( µ − e, T i ) ≤ . × − is used. For light flavor mesons, the prediction of BR( X → eµ ) is Table 1. Predictions on branching ratios of LFV decaysof vector mesons with d U = 3, d U = 3 . d U = 4,where CR( µ − e, T i ) ≤ . × − is used.Decay d U =3 d U =3.5 d U =4 ρ → eµ . × − . × − . × − ω → eµ . × − . × − . × − φ → eµ . × − . × − . × − J/ Ψ → eµ . × − . × − . × − Υ → eµ . × − . × − . × − much little, and it is impossible to observe the LFV processes of these mesonsin experiment. For heavy flavor mesons, the large prediction of BR( X → eµ ) isavailable for d U near 4. Especially, the prediction of BR(Υ → eµ ) is as large as O (10 − ), and it is very promising to be observed in experiment.In literatures, several stringent limits on LFV decays of vector mesons are derivedalready. A summary table of experimental bounds and corresponding theoreticalpredictions is presented in Tab.2. Assuming that a vector boson M i couples to µ ∓ e ± and e ∓ e ± , the authors of Ref. 26 deduce some upper bounds on the LFVdecay of mesons using the experimental constraint on the LFV process µ → e .Under a similar assumption that a vector meson M i couples to µ ∓ e ± and nucleon-nucleon, Ref. 27 and Ref. 28 study the LFV decays of vector mesons by takingaccount of the experimental constraint on µ − e conversion. From Tab.1 and Tab.2,it is easy to find that our predictions are compatible with those in literatures.Finally, the predictions of LFV decays of vector mesons in both our article andRef. 26, 27, 28 greatly depend on the experimental constraints on BR( l i → l j γ ),pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Lepton flavor violation decays of vector mesons in unparticle physics ? ρ → eµ − ≤ . × − ≤ . × − ω → eµ − ≤ . × − ≤ . × − φ → eµ ≤ . × − ≤ . × − ≤ . × − J/ Ψ → eµ < . × − ≤ . × − ≤ . × − Υ → eµ − ≤ . × − ≤ . × − BR( l i → l j ) and CR( µ − e ). Thus, more reliable predictions on LFV decays ofvector mesons depend on the new data from the experiment. In the future, theexpected sensitivities for BR ( µ → eγ ) would be of order 10 − .42 For BR ( τ → eγ )and BR ( τ → µγ ), it would be 10 − .43 For CR ( µ − e, T i ), it would be as low as10 − ∼ − .44 Then, the predictions of BR( X → eµ ) for vector mesons wouldbe more stringent.
4. Conclusions
Considering the constraint on µ − e conversion, we analyze the LFV decays of vectormesons in the framework of unparticle physics. In the scenario of the unparticlephysics, the predictions of branching ratios of LFV decays of vector mesons dependstrongly on the scale dimension d U . Supposing the unparticle couplings with the SMfermions are universal, the predictions of the branching ratios of the LFV decays ofvector mesons can reach the detective sensitivity in experiment in region of d U ≥
3, while the prediction of µ − e conversion rate in Ti nucleus is compatible withthe experimental upper limit. Although nonzero neutrino masses supported by theneutrino oscillation experiments imply the nonconservation of lepton flavors, it isvery important to directly search the LFV processes of charged lepton sector incolliders running now. The LFV process Υ → eµ is very promising to be observedin experiment. Acknowledgements
The work has been supported by the National Natural Science Foundation of China(NNSFC) with Grants No. 10975027.
References
1. Y. Fukuda et al. (Super Kamiokande Collab.), Phys. Rev. Lett ,1562(1998).2. Q. R. Ahmad et al. (SNO Collab.), Phys. Rev. Lett ,071301(2001).3. K. Eguchi et al. (Kamland Collab.), Phys. Rev. Lett ,021802(2003).4. J.C. Pati and A. Salam, Phys. Rev. D ,275(1974).5. H. Georgi and S. L. Glashow, Phys. Rev. Lett ,438(1974).6. P. Langacker, Phys. Rep ,185(1981).7. H. E. Haber and G. L. Kane, Phys. Rep ,75(1985). pril 30, 2018 20:19 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc.
8. C.-H. Chang, T.-F. Feng, Eur. Phys. J.C ,137(2000).9. K.-S. Sun, T.-F. Feng, T.-J. Gao, S.-M. Zhao, Nucl. Phys. B,DOI:10.1016/j.nuclphysb.2012.08.005.10. R. N. Mohapatra and J. C. Pati, Phys. Rev. D ,566(1975).11. R. N. Mohapatra and J. C. Pati, Phys. Rev. D ,2558(1975).12. G. Senjanovic and R. N. Mohapatra, Phys. Rev. D ,1502(1975).13. H. Georgi, Phys. Rev. Lett ,221601(2007).14. H. Georgi, Phys. Lett. B ,275(2007).15. T. M. Aliev, A. S. Cornell, N. Gaur, Phys. Lett. B ,27(2007).16. D. Choudhury, D.K. Ghosh, Mamta, Phys. Lett. B ,148(2008).17. G.-J. Ding, M.-L. Yan, Phys. Rev. D ,014005(2008).18. C.-D. Lu, W. Wang, Y.-M. Wang, Phys. Rev. D ,077701(2007).19. Wei.Zheng-Tao, Xu.Ye, Li.Xue-Qian, Eur. Phys. J. C ,593(2009).20. E.O. Iltan,Eur. Phys. J. C ,105(2008).21. E.O. Iltan, Mod. Phys. Lett. A ,3331(2008).22. Zuo-Hong. Li,Ying. Li,Hong-Xia. Xu, Phys. Lett. B ,150(2009).23. M. N. Achasov,et al. (SND Collaboration), Phys. Rev. D ,057102(2010).24. M. Ablikim et al. (BES Collaboration), Phys. Lett. B ,172(2004).25. W. Love et al. (CLEO Collaboration), Phys. Rev. Lett ,201601(2008).26. S. Nussinov, R.D. Peccei, X.M. Zhang, Phys. Rev. D ,016003(2000).27. T. Gutsche, J. Helo, S. Kovalenko, V.E. Lyubovitskij, Phys. Rev. D ,037702(2010).28. T. Gutsche, J. Helo, S. Kovalenko, V.E. Lyubovitskij, Phys. Rev. D ,115015(2011).29. T. Banks and A. Zaks, Nucl. Phys. B ,189(1982).30. K. Cheung, W.Y. Keung and T. C. Yuan, Phys. Rev. Lett ,051803(2007).31. T. Li, S.-M. Zhao, X.-Q. Li, Nucl. Phys. A ,125(2009).32. P. Ball, V.M. Braun, Phys. Rev. D ,2182(1996).33. T.-J. Gao, T.-F. Feng, X.-Q. Li, Z.-G. Si, S.-M. Zhao, Sci. China G ,1988(2010).34. M. Beneke, G. Buchalla, M. Neubert, C. T. Sachrajda, Nucl. Phys. B ,313(2000).35. S.-L. Chen, X.-G. He, X.-Q, Li, H.-C. Tsai, Z.-T. Wei,Eur. Phys. J.C ,899(2009).36. Z.-Q. Zhang, Phys. Rev. D ,034036(2010).37. Q. Wang,X.-H. Liu, Q. Zhao, hep-ph/1103.1095v1.38. H.-W. Ke,X.-Q. Li,Z.-T. Wei, X. Liu, Phys. Rev. D ,034023(2010).39. G. Mack,Commun. Math. Phys (1977)1.40. B. Grinstein, K.A. Intriligator, I.Z. Rothstein, Phys. Lett. B ,367(2008).41. R. Kitano, M. Koike, and Y. Okada, Phys. Rev. D66