The lepton flavor violating decays Z→ l i l j in the simplest little Higgs model
aa r X i v : . [ h e p - ph ] A p r The lepton flavor violating decays Z → l i l j in the simplest little Higgs model Xiaofang Han
Department of Physics, Yantai University, Yantai 264005, China
Abstract
In the simplest little Higgs model the new flavor-changing interactions between heavy neutrinosand the Standard Model leptons can generate contributions to some lepton flavor violating decaysof Z -boson at one-loop level, such as Z → τ ± µ ∓ , Z → τ ± e ∓ , and Z → µ ± e ∓ . We examine thedecay modes, and find that the branching ratios can reach 10 − for the three decays, which shouldbe accessible at the Giga Z option of the ILC. PACS numbers: 13.38.Dg, 12.60.-i, 11.30.Fs*) Email address: [email protected] . INTRODUCTION Little Higgs theory [1] has been proposed as an interesting solution to the hierarchy prob-lem. So far various realizations of the little Higgs symmetry structure have been proposed[2–5], which can be categorized generally into two classes [6]. One class use the productgroup, represented by the littlest Higgs model [3], in which the SM SU (2) L gauge group isfrom the diagonal breaking of two (or more) gauge groups. The other class use the simplegroup, represented by the simplest little Higgs model (SLHM) [4], in which a single largergauge group is broken down to the SM SU (2) L . The flavor sector of little Higgs modelsbased on product groups, notably the littlest Higgs model with T-parity (LHT) [5], hasbeen extensively studied [7]. Recently, some attentions have been paid to the flavor sectorof SLHM [8–10].The lepton flavor violating (LFV) decays of Z -boson can be a sensitive probe for newphysics because they are extremely suppressed in the SM but can be greatly enhanced innew physics models [11–13]. The experimental limits obtained at LEP [14] are BR ( Z → τ ± µ ∓ ) < . × − ,BR ( Z → τ ± e ∓ ) < . × − , (1) BR ( Z → µ ± e ∓ ) < . × − . The next generation Z factory can be realized in the Giga Z option of the InternationalLinear Collider (ILC) [15]. About 2 × Z events can be generated in an operational yearof 10 s of Giga Z . Thus the expected sensitivity of Giga Z to the LFV decays of Z -bosoncould reach [16] BR ( Z → τ ± µ ∓ ) ∼ κ × . × − ,BR ( Z → τ ± e ∓ ) ∼ κ × . × − , (2) BR ( Z → µ ± e ∓ ) ∼ . × − , with the factor κ ranging from 0.2 to 1.0. Therefore, Giga Z can offer an important oppor-tunity to probe the new physics via the LFV decays of Z -boson.The SLHM predicts the existence of heavy neutrinos, which have flavor-changing cou-plings with the SM leptons mediated respectively by the SM gauge boson W ± and the newheavy gauge boson X ± . These couplings can give great contributions to Z -boson decays2 → τ ± µ ∓ , Z → τ ± e ∓ , and Z → µ ± e ∓ at one-loop level. In this paper, we will calculatethe branching ratios of these decay modes, and compare the results with the sensitivity ofGiga Z and the present experimental bounds, respectively.This work is organized as follows. In Sec. II we recapitulate the SLHM. In Sec. III westudy respectively the decays Z → τ ± µ ∓ , Z → τ ± e ∓ and Z → µ ± e ∓ . Finally, we give ourconclusion in Sec. IV. II. SIMPLEST LITTLE HIGGS MODEL
The SLHM is based on [ SU (3) × U (1) X ] global symmetry [4]. The gauge symmetry SU (3) × U (1) X is broken down to the SM electroweak gauge group by two copies of scalarfields Φ and Φ , which are triplets under the SU (3) with aligned VEVs f and f . Theuneaten five pseudo-Goldstone bosons can be parameterized asΦ = e i t β Θ f , Φ = e − itβ Θ f , (3)where Θ = 1 f HH † + η √ , (4) f = p f + f and t β ≡ tan β = f /f . Under the SU (2) L SM gauge group, η is a realscalar, while H transforms as a doublet and can be identified as the SM Higgs doublet. Thekinetic term in the non-linear sigma model is L Φ = X j =1 , (cid:12)(cid:12)(cid:12)(cid:16) ∂ µ + igA aµ T a − i g x B xµ (cid:17) Φ j (cid:12)(cid:12)(cid:12) , (5)where g x = gt W / p − t W /
3, and t W = tan θ W with θ W being the electroweak mixing angle.As Φ and Φ develop their VEVs, the new heavy gauge bosons Z ′ , Y , Y † and X ± get3heir masses after eating five Goldstone bosons, M X = gf √ (cid:18) − v f (cid:19) ,M Z ′ = √ gf p − t W (cid:18) − − t W c W v f (cid:19) ,M Y = gf √ . (6)The gauged SU (3) symmetry promotes the SM fermion doublets into SU (3) triplets. Foreach generation of lepton, a heavy neutrino is added, whose mass is m N i = f s β λ iN . (7)Where i = 1 , , λ iN is the Yukawa coupling constant.After the EWSB the light and the heavy neutrino of the same family have the mixing,which is parameterized by δ v = − v √ ft β . The mixing angel δ v is experimentally constrainedto be small [17], and taken as a typical upper limit δ v < .
05 following the ref. [8]. Besides,there is family mixing as long as the Yukawa matrix of heavy neutrinos and that of leptonsare not aligned. This can induce the lepton flavor-changing interactions of charged currentsproportional to V ijℓ ¯ N Li γ µ X + µ ℓ Lj and δ v V ijℓ ¯ N Li γ µ W + µ ℓ Lj , where V ijℓ is the mixing matrix[6, 8, 9]. III. THE LFV DECAYS Z → τ ± µ ∓ , Z → τ ± e ∓ AND Z → µ ± e ∓ In the SLHM, the Feynman diagrams for Z → µ ± e ∓ can be depicted by the Fig. 1, andthe diagrams for Z → τ ± µ ∓ , Z → τ ± e ∓ are same as Fig.1, but replacing µ and e with thecorresponding final particles. For the ’t Hooft-Feynman gauge, the flavor-changing inter-actions between the heavy neutrino and lepton, mediated by the gauge bosons (Goldstonebosons) X ± ( x ± ) and W ± ( φ ± ), can contribute to these decays. The relevant Feynman rulescan be found in [8].The calculations of the loop diagrams in Fig. 1 are straightforward. Each loop diagramis composed of some scalar loop functions [18] which are calculated by using LoopTools [19].The analytic expressions from our calculation are presented in Appendix A.The SM input parameters relevant in our study are taken as ref. [20]. The free SLHMparameters involved are f, t β , the heavy neutrino mass m N i ( i = 1 , , N i X ( W ) ν j µe (a) Z ν i X ( W ) N j µe (b) Z N i X ( W ) N j µe (c) Z X ( W ) N i X ( W ) µe (d) Z N i x ( φ ) ν j µe (e) Z ν i x ( φ ) N j µe (f) Z N i x ( φ ) N j µe (g) Z x ( φ ) N i x ( φ ) µe (h) Z X ( W ) N i x ( φ ) µe (i) Z x ( φ ) N i X ( W ) µe (j) Z X ( W ) N i e µe (k) Z X ( W ) N i µ µe (l) Z x ( φ ) N i e µe (m) Z x ( φ ) N i µ µe (n) FIG. 1: Feynman diagrams for Z → µ + e − in the SLHM. matrix V ℓ which can be parameterized with standard form. To simply our calculations, wetake the parameters [21] s = √ . , s = √ . , s = 1 √ , δ = 65 ◦ , (8)which is consistent with the experimental constraints on the PMNS matrix [22], and δ is taken to be equal to the CKM phase. To satisfy the present experimental bounds of Br ( µ → eγ ) and Br ( µ → eee ), the mass splitting of the first and the second heavy neutrinosmust be very small [8]. So in this paper we will take m N = m N = m = 400 GeV and m N = m in the range of 500 GeV-3000 GeV. Ref. [4] shows that the LEP-II data requires f > t β for f = 2 TeV, f = 4 TeV and f = 5 . Z → τ ± µ ∓ , Z → τ ± e ∓ -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=2TeVm (GeV) B r ( Z → t m ) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=4TeVm (GeV) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=5.6TeVm (GeV) FIG. 2: The branching ratios of Z → τ ± µ ∓ versus m . -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=2TeVm (GeV) B r ( Z → t e ) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=4TeVm (GeV) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=5.6TeVm (GeV) FIG. 3: The branching ratios of Z → τ ± e ∓ versus m . and Z → µ ± e ∓ versus m for f = 2 TeV, f = 4 TeV and f = 5 . m − m , which increasesas m gets large since we have fixed the value of m . Besides, the branching ratios dropas the scale f or t β get large, and the reason is that the lepton flavor-changing couplings¯ N Li γ µ W + µ ℓ Lj and ¯ N i φ + ℓ j are proportional to δ v = − v √ ft β .Fig. 2, Fig. 3 and Fig. 4 show the branching ratios of Z → τ ± µ ∓ , Z → τ ± e ∓ and Z → µ ± e ∓ are below the present experimental upper bounds, respectively. However, theratios can be enhanced to reach the sensitivity of the Giga Z . For f = 2 TeV, t β = 4 and m = 2 TeV, the branching ratios can reach 10 − for Z → τ ± µ ∓ , Z → τ ± e ∓ and Z → µ ± e ∓ ,which exceed much the sensitivity of Giga Z . In the LHT, all the three ratios can reach 10 − [12]. Therefore, the LFV decays of Z -boson may be accessible at Giga Z , and thus may serveas a probe of the little Higgs models. 6 -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=2TeVm (GeV) B r ( Z → m e ) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=4TeVm (GeV) -11 -10 -9 -8 -7 -6 -5
500 1000 1500 2000 2500 3000 t b =1t b =2t b =4t b =10 sensitivity of GigaZupper bound from LEP f=5.6TeVm (GeV) FIG. 4: The branching ratios of Z → µ ± e ∓ versus m . IV. CONCLUSION
In the framework of the simplest little Higgs model, we studied the LFV decays Z → τ ± µ ∓ , Z → τ ± e ∓ and Z → µ ± e ∓ . In the parameter space allowed by current experi-ments, the branching ratios of the three decays can exceed respectively much the sensitivityof Giga Z , which should be accessible at the Giga Z option of the ILC. Therefore, the mea-surement of these rare decays at the Giga Z may serve as a probe of the simplest little Higgsmodel. Acknowledgment
This work was supported in part by the National Natural Science Foundation of China(NNSFC) under grant No. 11005089, and by the Foundation of Yantai University underGrant Nos. WL10B24 and WL09B31.
Appendix A: The effective coupling of Zµ + e − Here we take the effective coupling of Zµ + e − for example. The other two couplings Zτ + µ − and Zτ + e − can be obtained via some corresponding replacement of the analytic7xpressions for Zµ + e − . The effective coupling of Zµ + e − is given byΓ αZµe = Γ αV F F (cid:2) X ( W ) , N i , ν j (cid:3) + Γ αV F F (cid:2) X ( W ) , ν i , N j (cid:3) + Γ αV F F (cid:2) X ( W ) , N i , N j (cid:3) +Γ αF V V (cid:2) N i , X ( W ) , X ( W ) (cid:3) + Γ αSF F (cid:2) x ( φ ) , N i , ν j (cid:3) + Γ αSF F (cid:2) x ( φ ) , ν i , N j (cid:3) +Γ αSF F (cid:2) x ( φ ) , N i , N j (cid:3) + Γ αF SS (cid:2) N i , x ( φ ) , x ( φ ) (cid:3) + Γ αF V S (cid:2) N i , X ( W ) , x ( φ ) (cid:3) +Γ αF SV (cid:2) N i , x ( φ ) , X ( W ) (cid:3) + Γ αself ( k ) (cid:2) X ( W ) , N i (cid:3) + Γ αself ( l ) (cid:2) X ( W ) , N i (cid:3) +Γ αself ( m ) (cid:2) x ( φ ) , N i (cid:3) + Γ αself ( n ) (cid:2) x ( φ ) , N i (cid:3) , (A1)where the particles in the square brackets represent the particles which contribute to thevertex, and Γ αself ( k − n ) correspond to the vertexes in Fig. 1( k − n ). The self-energy and vertexcontributions in the above equation are given by Γ αV F F = i π (cid:2) ( d Z fR P L + c Z fL P R )( − C σρ γ σ γ α γ ρ − γ α )( c P L + d P R ) − q/ e + q/ µ ) γ α C β γ β ( c c Z fL P L + d d Z fR P R ) + 4 m F ( c d Z fR P L + c d Z fL P R ) C α + 4 m F ( c d Z fL P L + c d Z fR P R ) C α + 2 m F C ( d Z fL P L + c Z fR P R )(2( q e + q µ ) α − m F γ α )( c P L + d P R ) (cid:3) ( q e , q µ , m F , m V , m F ) , (A2)Γ αF V V = ig V V V π ( d P L + c P R ) (cid:8) − C αβ γ β + γ α − C β γ β ( q e + q µ ) α + 2(4 m F − q/ µ ) C α +(4 m F − q/ µ )( q e + q µ ) α C − (cid:2) C σρ g σρ − (cid:3) γ α − C β γ β ( q/ µ + m F ) γ α − p/ Z C β γ β γ α − p/ Z ( q/ µ + m F ) γ α C − (cid:2) C σρ g σρ − (cid:3) γ α + C β γ α γ β ( p/ Z − q/ e − q/ µ ) − γ α ( q/ µ + m F ) C β γ β + γ α ( q/ µ + m F )( p/ Z − q/ e − q/ µ ) C (cid:9) ( c P L + d P R )( q µ , q e , m V , m F , m V ) , (A3)Γ αSF F = i π (cid:2) C σρ γ σ γ α γ ρ ( a b Z fR P L + a b Z fL P R ) + 12 γ α ( a b Z fR P L + a b Z fL P R )+ C β γ β γ α ( a Z fL P L + b Z fR P R )( q/ e + q/ µ + m F )( a P L + b P R )+ m F γ α ( b b Z fL P L + a a Z fR P R ) C β γ β + m F γ α ( b Z fL P L + a Z fR P R )( q/ e + q/ µ + m F )( a P L + b P R ) C (cid:3) ( q e , q µ , m F , m S , m F ) , (A4) αF SS = − ig V SS π (cid:8) − C αβ γ β ( a b P L + a b P R ) − ( q e + q µ ) α C β γ β ( a b P L + a b P R )+ (cid:2) − C α − ( q e + q µ ) α C (cid:3) q/ µ ( a b P L + a b P R ) + m F (cid:2) − C α − ( q e + q µ ) α C (cid:3) ( a a P L + b b P R ) (cid:9) ( q µ , q e , m S , m F , m S ) , (A5)Γ αF V S = − ig V V S π γ α ( c P L + d P R ) (cid:2) C β γ β + ( q/ µ + m F ) C (cid:3) × ( a P L + b P R )( q µ , q e , m S , m F , m V ) , (A6)Γ αF SV = ig V V S π ( a P L + b P R ) (cid:2) C β γ β + ( q/ µ + m F ) C (cid:3) γ α × ( c P L + d P R )( q µ , q e , m V , m F , m S ) , (A7)Γ αself ( k ) = − ig π c W ( q µ − m e ) γ α (cid:2) ( −
12 + s W ) P L + s W P R (cid:3) ( q/ µ + m e ) (cid:2) (2 B β γ β + (2 B − q/ µ )( c c P L + d d P R ) − m F (2 B − c d P L + c d P R ) (cid:3) ( q µ , m V , m F ) , (A8)Γ αself ( l ) = − ig π c W ( p e − m µ ) (cid:2) (2 B β γ β + (2 B − p/ e )( c c P L + d d P R ) − m F (2 B − c d P L + c d P R ) (cid:3) ( p/ e + m µ ) γ α (cid:2) ( −
12 + s W ) P L + s W P R (cid:3) ( p e , m V , m F ) , (A9)Γ αself ( m ) = ig π c W ( q µ − m e ) γ α (cid:2) ( −
12 + s W ) P L + s W P R (cid:3) ( q/ µ + m e ) (cid:2) ( B β γ β + q/ µ B )( a b P L + a b P R ) + m F B ( a a P L + b b P R ) (cid:3) ( q µ , m S , m F ) , (A10)Γ αself ( n ) = ig π c W ( p e − m µ ) (cid:2) ( B β γ β + p/ e B )( a b P L + a b P R ) + m F B ( a a P L + b b P R ) (cid:3) ( p/ e + m µ ) γ α (cid:2) ( −
12 + s W ) P L + s W P R (cid:3) ( p e , m S , m F ) , (A11) where q µ = − p µ , q e = − p e and P L,R = (1 ∓ γ ) /
2. The functions B and C are 2- and3-point Feynman integrals [19], and their functional dependence is indicated in the bracketfollowing them. The tensor loop functions can be expanded as the scalar functions [19]. Inour calculation the contraction of Lorentz indices is performed numerically. The parametersappearing above are from V ¯ ef : iγ µ ( c P L + d P R ) , V ¯ f µ : iγ µ ( c P L + d P R ) ,S ¯ ef : a P L + b P R , S ¯ f µ : a P L + b P R ,ZS + S − : ig V SS ( p µS + − p µS − ) , ZV + S − : g V V S g µν ,Z ρ V + µ V − ν : − ig V V V [( p V + − p V − ) ρ g µν + ( p Z − p V + ) ν g µρ + ( p V − − p Z ) µ g νρ ] ,Z ¯ f f : iγ µ ( Z fL P L + Z fR P R ) , where V represents gauge bosons and S represents scalar particles. These couplings representthe seven different classes of vertices involved in our calculation. In each class of vertices,9he parameters a , b , a , b , c , d , c , d , g V SS , g V V S , g V V V , Z fL and Z fR take differentvalues for different concrete coupling. The analytic expressions of these parameters can befound in [8]. [1] N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B , 232 (2001); N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire and J. G. Wacker, JHEP , 021(2002).[2] D. E. Kaplan and M. Schmaltz, JHEP , 039 (2003); I. Low, W. Skiba, and D. Smith,Phys. Rev. D , 072001 (2002); S. Chang and J. G. Wacker, Phys. Rev. D , 035002 (2004);T. Gregoire, D. R. Smith, and J. G. Wacker, Phys. Rev. D , 115008 (2004); W. Skiba andJ. Terning, Phys. Rev. D , 075001 (2003); S. Chang, JHEP , 057 (2003); H. Cai, H.-C.Cheng, and J. Terning, JHEP , 045 (2009); A. Freitas, P. Schwaller, and D. Wyler,JHEP , 027 (2009).[3] N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP , 034 (2002).[4] M. Schmaltz, JHEP , 056 (2004).[5] H. C. Cheng, I. Low, JHEP , 051 (2003); JHEP 0408, 061 (2004); H. C. Cheng, I. Lowand L. T. Wang, Phys. Rev. D , 055001 (2006); J. Hubisz and P. Meade, Phys. Rev. D ,035016 (2005).[6] T. Han, H. E. Logan and L. T. Wang, JHEP , 099 (2006).[7] M. Blanke, A. J. Buras, A. Poschenrieder, S. Recksiegel, C. Tarantino, S. Uhlig, and A. Weiler,JHEP , 062 (2006); J. Hubisz, S. J. Lee, and G. Paz, JHEP ,041 (2006); T. Goto,Y. Okada, and Y. Yamamoto, Phys. Lett. B , 378 (2009); A. Paul, I. I. Bigi, S. Recksiegel,Phys. Rev. D , 094006 (2010); F. Penunuri, F. Larios, Phys. Rev. D , 015013 (2009);X.-F. Han, L. Wang, J. M. Yang, Phys. Rev. D , 075017 (2008); Phys. Rev. D , 015018(2009); S. Fajfer, J. F. Kamenik, JHEP , 074 (2007); C.-H. Chen, C.-Q. Geng, T.-C.Yuan, Phys. Lett. B , 50-57 (2007); A. Belyaev, C.-R. Chen, K. Tobe, C.-P. Yuan, Phys.Rev. D , 115020 (2006).[8] F. del Aguila, J. I. Illana, M. D. Jenkins, JHEP , 080 (2011).[9] J. I. Illana, M. D. Jenkins, Acta Phys. Polon. B , 3143 (2009); F. d. Aguila, J. I. Illana, M.D. Jenkins, Nucl. Phys. Proc. Suppl. , 158-163 (2010).
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