The branching ratio R b in the littlest Higgs model

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1 The ranching ratio R in the littlest Higgs model Chongxing Yue and Wei Wang Department of Physics, Liaoning Normal University, Dalian 11609, China Feruary 1, 008 arxiv:hep-ph/040114v1 7 Jan 004 Astract In the context of the littlest Higgs(LH) model, we study the contriutions of the new particles to the ranching ratio R. We find that the contriutions mainly dependent on the free parameters f, c and x L. The precision measurement value of R gives severe constraints on these free parameters. PACS numer: 1.60.Cn, Cp, 1.15.Lk cxyue@lnnu.edu.cn 1

2 I. Introduction It is well know that most of the electroweak olique and QCD corrections to the Z ranching ratio R cancel etween numerator and denominator, R is very sensitive to the new physics(np) eyond the standard model(sm). The precision experimental value of R may give a severe constraint on the NP[1]. Thus, it is very important to study the Z process in extensions of the SM and pursue the resulting implications. Little Higgs models[,3,4] provide a new approach to solve the hierarchy etween the TeV scale of possile NP and the electroweak scale, ν = 46GeV = ( G F ) 1/. In these models, at least two interactions are needed to explicitly reak all of the gloal symmetries, which forid quadratic divergences in the Higgs mass at one-loop. Electroweak symmetry reaking(ewsb) is triggered y a Coleman-Weinerg potential, which is generated y integrating out the heavy degrees of freedom. In this kind of models, the Higgs oson is a pseudo-goldstone oson of a gloal symmetry which is spontaneously roken at some higher scale f y an vacuum expectation value(vev) and thus is naturally light. A general feature of this kind of models is that the cancellation of the quadratic divergences is realized etween particles of the same statistics. Little Higgs models are weakly interaction models, which contain extra gauge osons, new scalars and fermions, apart from the SM particles. These new particles might produce characteristic signatures at the present and future collider experiments[5,6,7]. Since the extra gauge osons can mix with the SM gauge osons W and Z, the masses of the SM gauge osons W and Z and their couplings to the SM particles are modified from those in the SM at the order of ν f. Thus, the precision measurement data can give severe constraints on this kind of models[5,8,9]. Aim of this paper is to consider the Z ranching ratio R in the context of the littlest Higgs(LH) model[] and see whether the new particles predicted y the LH model can give significant contriutions to R. We find that, compare the calculated value of R with the experimental measured value, the precision data can give severe constraint on the free parameters of the LH model. The LH model has een extensively descried in literature. However, in order to

3 clarify notation which is relevant to our calculation, we will simply review the LH model in section II. In section III, we discuss the effects of the new gauge osons on the ranching ratio R. We calculate the contriutions of the top quark t and vector-like quark T to R via the couplings Wt, WT, W t and W T in section IV. The contriutions of the new scalars to R are studied in section V. Discussions and conclusions are given in section VI. II. Littlest Higgs model The LH model[] is emedded into a non-linear σ model with the coset space of SU(5)/SO(5). At the scale Λ s 4πf, the gloal SU(5) symmetry is roken into its sugroup SO(5) via a VEV of order f, resulting in 14 Goldstone osons. The effective field theory of these Goldstone osons is parameterized y a non-linear σ model with gauge symmetry [SU() U(1)], spontaneously roken down to its diagonal sugroup SU() U(1), identified as the SM electroweak gauge group. Four of these Goldstone osons are eaten y the roken gauge generators, leaving 10 states that transform under the SM gauge group as a doulet H and a triplet Φ. A new charge /3 quark T is also needed to cancel the divergences from the top quark loop. The effective non-linear Lagrangian invariant under the local gauge group [SU() 1 U(1) 1 ] [SU() U(1) ], which can e written as[5,9]: eff = G + F + Y + Σ V CW, (1) where G consists of the pure gauge terms, which can give the 3 and 4 particle interactions among the SU() gauge osons and the couplings of the U(1) gauge osons to the SU() gauge osons. The fermion kinetic term F can give the couplings of the gauge osons to fermions. The couplings of the scalars H and Φ to fermions can e derived from the Yukawa interaction term Y. In the LH model, the gloal symmetry prevents the appearance of a Higgs potential at tree level. The effective Higgs potential, the Coleman-Weinerg potential V CW [10], is generated at one-loop and higher orders due to interactions with gauge osons and fermions, which can induce to EWSB y driving the Higgs mass squared parameter negative. Σ consists of the σ model terms of the LH 3

4 model. The scalar Σ field can e written as: Σ = e iπ/f Σ 0 e iπt /f = e iπ/f Σ 0 () with Σ 0 f which generates masses and mixing etween the gauge osons. The ten pseudo-goldstone osons can e parameterized as: 0 H + / Φ + Π = H/ 0 H /. (3) Φ H T / 0 Where H is identified as the SM Higgs doulet, H = (H +, H 0 ), and Φ is a complex SU() triplet with hypercharge Y =, Φ = Φ++ Φ + / Φ + / Φ 0 The kinetic terms of the scalar Σ field are given y where the covariant derivative of the Σ field is defined as:. (4) Σ = f 8 Tr{(D µσ)(d µ Σ) + }, (5) D µ Σ = µ Σ i j [g j W a j (Q a jσ + ΣQ at j ) + g jb j (Y j Σ + ΣY T j )], (6) where g j, g j are the gauge coupling constants, W j, B j are the gauge osons, Q a j and Y j are the generators of gauge transformations. The gauge oson mass eigen-states are given y with the cosines of two mixing angles, W = sw 1 + cw, W = cw 1 + sw, B = s B 1 + c B, B = c B 1 + s B (7) c = g 1 g 1 + g, c = g 1. (8) g 1 + g The SM gauge coupling constants are g = g 1 s = g c and g = g 1 s = g c. 4

5 From the effective non-linear Lagrangian L, one can derive the mass and coupling expressions of the gauge osons, scalars and the fermions, which have een extensively discussed in Refs.[5,9]. The mass spectrum of the LH model and the coupling forms which are related our calculations are summarized in appendix A, B and C, respectively. III. New gauge osons and the Z ranching ratio R 1. Corrections of new physics to the Z ranching ratio R In general, the effective Z vertex can e written as: [gl Lγ µ L + gr Rγ µ R ] Z µ (9) with the form factors g L and g R : g L g R = g,sm L + δg L = e S W C W ( S W ) + δg L, = g,sm R + δg R = e S W C W ( 1 3 S W ) + δg R. (10) Where S W = sin θ W, θ W is the Weinerg angle. δg L and δg R represent the corrections of NP to the left-handed and right-handed Z couplings, respectively. Certainly, the corrections of NP to the Z couplings g L and g R may give rise to one additional form factor, proportional to σ µν q ν. However, its contriutions to R are very small and can e ignored[1]. The ranching ratio R can e written as: R = Γ Γ h = Γ 3Γ + Γ c. (11) Here Γ c is the width of the process Z cc. The partial decay width, Γ q, of the Z qq decay(q = u, d, c, s, and ) is given as[11]: with Γ 0 = G F m 3 Z 4 αs. The factor π π Γ q = 6Γ 0 (1 + α s π )[(gq L ) + (g q R ) ], (1) contain contriutions from the find state gluons and photons. In aove equation, the masses of the final quarks are assumed to e negligile. 5

6 Since the ranching ratio R is the ratio etween two hadronic widths, R is almost independent of the EW olique and QCD corrections ecause of the near cancellation of these corrections etween the numerator and the denominator. The remaining ones are asored in the definition of the renormalized coupling parameters α and S W, up to terms of high order in the electroweak corrections[1]. Thus, R is very sensitive to the NP eyond the SM. The correction of NP to R can e written as: δr = R R SM = ΓSM Γ SM h + δγ ΓSM + δγ h Γ SM h ( δγ δγ h )R SM Γ Γ h = R SM { (g L δg L + g R δg R ) 4(gc L δgc L + gc R δgc R ) + 6(g L δg L + g R δg R ) }. (13) (gl ) + (gr ) [(gl c ) + (gr c ) ] + 3[(gL ) + (gr ) ] In aove equation, we have neglected the terms of O[(δg q L,R ) ]. In the next susection, we will study the corrections of the new gauge osons predicted y the LH model to the ranching ratio R.. The extra gauge osons and the ranching ratio R The LH model predicts the existence of the extra gauge osons, such as W, Z and B. These new particles can generate corrections to the ranching ratio R via mixing with the SM gauge osons and the coupling to the SM fermions. The corrections to R mainly come from three sources: (1) the modifications of the relations etween the SM parameters and the precision electroweak input parameters, which come from the mixing of the heavy W oson to the couplings of the charge current and from the contriutions of the current to the equations of motion of the heavy gauge osons, () the correction terms to the Z couplings g L and g R coming from the mixing etween the extra gauge oson Z and the SM gauge oson Z, (3) the neutral gauge osons Z exchange and B exchange. In the LH model, the relation etween the Fermi coupling constant G F, the gauge oson Z mass m Z and the fine structure coupling constant α(m Z ) can e written as[9]: So we have G F = πα m Z S W C W e S W C W [1 g c G F s (c s ) ν f + c4ν f 5 4 (c s ) ν f]. (14) 8G F m Z =. (15) 1 g c G F s (c s ) ν + c f 4 ν 5 f 4 (c s ) ν f 6

7 In our numerical calculations, we will take G F = GeV, m Z = GeV and m t = 174.3GeV [13] as input parameters and use them to represent the other SM parameters. g q,sm L Due to the mixing etween the gauge osons Z and Z, the tree-level Zqq couplings and g q,sm R receive corrections at the order of ν f : δg q i,1 L = δg q i,1 R = δg q j,1 L = δg q j,1 R = e ν (c s ) + 5 S W C W f [c 4 6 (c s )( c )], (16) e ν S W C W f [5 3 (c s )( c )], (17) e ν (c s ) + 5 S W C W f [ c 4 6 (c s )( c )], (18) e ν S W C W f [5 3 (c s )( c )], (19) where q i and q j represent the down-type quarks(d, s, ) and the up-type quarks(c, s),respectively f=1tev f=tev f=3tev f=4tev R LG R exp Figure 1: The ranching ratio R LG parameter c = 1, f = 1TeV, TeV, 3TeV and 4TeV. c as a function of the mixing parameter c for the mixing 7

8 Using the same method as calculating the contriutions of the topcolor gauge osons to R [14,15], we can give the corrections of the neutral gauge oson Z exchange and B exchange to the Zqq couplings g q L and gq R : δg q i, L = δg q j, L = δg q i,3 L = δg q i,3 R = δg q j,3 L = δg q j,3 R = e c m Z 4π SWs e c m Z 4π SW s M Z ln M Z m Z ln M Z MZ m Z e 54π CWs c [1 5 1 c ] m Z g q i,sm L, δg q i, R = 0, (0) g q j,sm L, δg q j, R = 0, (1) M B ln M B e 7π CWs c [ c ] m Z e 54π CWs c [1 5 1 c ] m Z 8e 7π CWs c [1 5 1 c ] m Z m Z M B ln M B m Z M B ln M B m Z M B ln M B m Z g q i,sm R, () g q i,sm R, (3) g q j,sm R, (4) g q j,sm R. (5) Where M Z and M B are the masses of the gauge oson Z and the heavy photon B, respectively, which have een listed in appendix A. In aove equations, we have used the expressions of the Z qq and B qq couplings given in appendix B. Adding all the corrections together, we otain the total corrections of extra gauge osons to R : δg,g L = δg,1 L + δg, L + δg,3 L, δg c,g L = δg c,1 L + δgc, L + δgc,3 L, δg,g R = δg,1 R + δg,3 R, (6) δgc,g R = δgc,1 R + δgc, R + δgc,3 R. (7) Plugging Eqs.(15)-(7) into Eq.(13), we can otain the relative correction δrlg R SM y the new gauge osons. In our calculation, we have taken R LG 0.157, and R exp = R SM + δr LG given, R SM = = ± [16]. Our numerical results are summarized in Fig.1 and Fig., in which we have used the horizontal solid line to denote the central value of R exp and dotted lines to show the 1σ and σ ounds. In Fig.1(Fig.) we plot R LG a function of the mixing parameter c(c ) for c = 1 (c = 1 ) and the scale parameter f = 1TeV (solid line), TeV (dashed line), 3TeV (dotted line) and 4TeV (dotted-dashed line). From Fig.1 and Fig., one can see that the contriutions of the new gauge osons to R decrease as the scale parameter f increasing. For c = 1, R LG as is insensitive to 8

9 the mixing parameter c and the value of δr LG is very small. For c = 1, the new gauge osons decrease the value of the ranching ratio R for c > 0.7 and f > 1TeV. To make the predicted R value satisfy the precision experimental value in σ ound, we should have 0.57 < c < 0.73 for f = 1TeV. For c = 1, f > TeV, the predicted value of R is consistent with the precision experimental value R exp allowed range of the mixing parameter c. within σ ound in most of the R LG R exp c' f=1tev f=tev f=3tev f=4tev Figure : The ranching ratio R LG as a function of the mixing parameter c for c = 1 and f = 1TeV, TeV, 3TeV and 4TeV. From aove discussions, we can see that the corrections of new gauge osons to R can e divided into two parts: one part is the tree-level corrections coming from the shift in the Z couplings to quarks and the modifications of the relations etween the SM parameters and the precision electroweak input parameters and the second part is the one-loop corrections coming from the neutral gauge osons Z exchange and B exchange. To compare the tree-level corrections with the one-loop corrections, we plot R = δr 1 loop /δr tree level as a function of the mixing parameter c for f = TeV and c = 1 in Fig.3. One can 9

10 see from Fig.3 that the one-loop contriution is smaller than the tree-level contriution at least y two orders of magnitude in all of the parameter space R ( 10-3 ) c' Figure 3: The relative correction R as a function of the mixing parameter c for f = TeV and c = 1. IV. The corrections of the top quark t and vector-like quark T to R It is well known that R is almost independent of the EW olique and QCD corrections. However, for Z couplings, there is an important correction coming from the top triangle diagrams, which can not e ignored. They can generate significant m t enhanced contriutions to R [17]. Furthermore, the extra top quark predicted y NP can also produce significant corrections to R at one-loop[1,18]. In this section, we will calculate the corrections of the top quark t and vector-like quark T to R via the couplings Wt, WT, W t and WT. The relevant Feynman diagrams are shown in Fig.4. Since the gauge osons W and W can only couple to the left handed quark s, t, T and, the top and vector-like top triangle loops have no contriutions to the right-handed 10

11 Z coupling gr. If we assume that the mass of the ottom quark is approximately equal to zero, then the corrections to the Z coupling g L generated y the Wt and W t couplings can e written as: with Lt = ( e ){ α S W C W 4πSW δg,1 [F 1 (x t ) + c s F 1(x t )] + 3αC W [F 8πSW (x t ) + c s F (x t )]} (8) F 1 (x) = gt L ) [x(x (x 1) ln x + x x 1 ] + x gt R[ (x 1) ln x x ], (9) x 1 x F (x) = (x 1) ln x x x 1, (30) where x t = ( mt m W ) and x t = ( mt M W ). In aove equation, we have neglected the interference effects etween gauge osons W and W. Ø Ì Ï Ï ¼ Ø Ì Ï Ï ¼ Ï Ï ¼ Ø Ì µ µ Figure 4: Feynman diagrams for the contriutions of the top quark t and vector-like quark T to the Z vertex. In the LH model, due to the mixing etween the top quark t and the vector like quark T, the tree-level Ztt couplings receive corrections at the order of ν f, which also have contriutions to the Z coupling g L : Lt = ( e α ) S W C W 16πSW δg, ( ν x L f )[x t( 4 x t 1 log x t) + c s x t ( 4 x t 1 log x t )]. (31) 11

12 R LT c x L =0. x L =0.5 x L =0.8 Figure 5: The ranching ratio R LT and three values of the scale parameter x L. as a function of the mixing parameter c for f = TeV The mixing angle parameter etween the SM top quark t and the vector-like quark T is defined as x L = λ 1, in which λ λ 1 and λ are the coupling parameters. 1 +λ The contriutions of the vector like top quark T to R via the couplings WT and W T can e written as: e Lt = ( ) ν x L S W C W f { α 4πSW δg,3 with [F 3 (x T ) + c s F 3(x T )] + 3αC W [F 8πSW (x T ) + c s F (x T )]} (3) F 3 (x) = gt L ) [x(x (x 1) ln x + x x 1 ] + x gt R[ (x 1) ln x x ], (33) x 1 where x T = ( M T m W ) and x T = ( M T M W ). The t T contriutions can e given y e α Lt = ( ) ( νx L S W C W 4πSW f )[ 1 x T ( x T x t x T 1 log x T x tx T x T ( x T x t x T 1 log x T δg,4 x t x t 1 log x t) x t x t 1 log x t)]. (34) 1

13 Being CKM suppression, the contriutions of the top quark t and vector-like quark T to the couplings of the gauge oson Z to other quarks(u, c, d, s) are very small, which can e ignored. Then Eq.(13) should e changed as, for calculating the contriutions of the quarks t and T: δr = R SM { g L δg L + g R δg R (gl) + (gr) gl δg L + g R δg R }. (35) [(gl) c + (gr) c ] + 3[(gL) + (gr) ] f=1tev f=tev f=3tev f=4tev R LT R exp Figure 6: The ranching ratio R LT as a function of the mixing parameter x L for c = 1 and four values of the scale parameter f. x L In Fig.5 we plot the ranching ratio R LT = R SM + δr t + δr T as a function of the mixing parameter c for f = TeV and three values of the mixing parameter x L. One can see from Fig.5 that the corrections of the top quark t and vector-like quark T to R are not sensitive to the value of the flavor mixing parameter c, while are strongly dependent on the mixing parameter x L. This is ecause the mass M W of the heavy gauge oson W suppresses the contriutions of the t and T quarks to R. To see the effects of x L varying on R, we plot R LT as a function of x L for c = 1, and four values of the scale 13

14 parameter f in Fig.6. One can see from Fig.6 that the top quark t and vector-like quark T can generate negative corrections to R for f 1TeV and x L 0.5, while they can give positive corrections to R for f 4TeV and x L V. The contriutions of scalars to the ranching ratio R Ø Ì Ø Ì µ µ Ø Ì Ø Ì Ø Ì µ µ Figure 7: Feynman diagrams for the contriutions of the charged scalars Φ ± to the Z vertex via the couplings Φt and ΦT. Since the douly charged scalars can not couple to the SM fermions, so they have no contriutions to R. The singly charged scalars Φ ± can give contriutions to the ranching ratio R via the couplings Φt and ΦT. The relevant Feynman diagrams for the corrections of the charged scalars Φ ± to the Z couplings g L and g R Fig.7. are shown in Using the Feynman rules given in appendix C and other Feynman rules, we can give: δg,s R = 0, δg,s L = e S W C W m t 3π ν (ν f 4ν ν )[A t + x L A T ], (36) 1 x L 14

15 with A q = g,sm L B 1 ( P, m q, M Φ ) + g t,sm R [C 4 (P, k, M Φ, m q, m q ) + B 0 ( k, m f, m q ) M ΦC 0(P, k, M Φ, m q, m q )] + m tg t,sm L C 0(P, k, M Φ, m q, m q ) +s W C 4( P, k, m q, M Φ, M Φ ), (37) where q represents the SM top quark t or the vector-like quark T. B i, C i and C ij are the standard Feynman integrals[19], in which the variale P is the momentum of quark, k is the momentum of the gauge oson Z R SM R LS x L =0. x L =0.5 x L = f Figure 8: The ranching ratio R LS of the mixing parameter x L. as a function of the scale parameter f for three values Certainly, the neutral scalars H 0 and Φ 0 can also generate corrections to the Z couplings g L and g R via the couplings H 0 and Φ 0. However, compared with the charged scalar contriutions, the neutral scalar contriutions are suppressed at least y the factor m and thus can e ignored. In order to get a positive definite mass M m Φ of t the triplet scalars, we should have that the value of the ratio of the scalar triplet VEV 15

16 ν to the scalar doulet VEV ν is smaller than ν. To simply our calculation, we assume 4f ν = 1 ν ν 5 f. In this case, the triplet scalar mass M Φ given in appendix A can e written as: M Φ = 10m H f /ν, which m H is the SM Higg mass R SM x L =0. x L =0.5 x L =0.8 R LS Figure 9: The ranching ratio R LS as a function of ν ν for f = 3TeV and x L = 0., 0.5 and 0.8. In Fig.8 we plot the ranching ratio R LS = R SM + δr LS as a function of the scale parameter f for x L = 0.(solid line), 0.5(dashed line), and 0.8(dotted line), in which we have taken the SM Higgs mass m H = 10GeV. We can see from Fig.8 that the charged scalars Φ ± generate the negative corrections to the ranching ratio R. The negative corrections increase as the scale parameter f decreasing and the mixing parameter x L increasing. For the parameter f, the corrections of the charged scalars to R go to zero. However, the varying value of R is very small and is smaller than that generated y the new gauge osons, the top quark t and vector-like quark T in most of the parameter space. To see the effects of varying the triplet scalar VEV ν on the ranching ratio R, we 16

17 take f = 3TeV, which means ν ν < ν 4f = The RLS is plotted in Fig.9 as a function of ν ν for f = 3TeV and three values of the mixing parameter x L. From Fig.9 we can see that the contriutions of the charged scalars to R decrease as the value of the ratio ν ν increasing for the fixed value of the parameter f and ν < ν. If we assume that the value ν 4f of the ratio ν ν ν goes to, then the correction value goes to zero. 4f VI. Discussions and conclusions The LH model predicts the existence of several scalars, new gauge osons, and vectorlike quark T. These new particles can generate corrections to the ranching ratio R. Thus, the predicted value of R can e written as R LH = R SM + δr LG + δr LT + δr LS the LH model. So, using the experimental value R exp, we might give the constraints on the free parameters of the LH model. in f=1tev f=tev f=3tev f=4tev R LH R exp Figure 10: The predicted value of R LH parameter c for four values of the scale parameter f. c' in the LH model as a function of the mixing From aove discussions one can see that the correction effects of the new particles 17

18 predicted y the LH model to the ranching ratio R decrease as the scale parameter f increasing. The charged scalars generate the negative correction to R in all of the parameter space. The correction value increases as the mixing parameter x L increasing, which is very small. The contriutions of the top quark t and the vector-like T are related to the parameters c and x L. However, they are insensitive to the parameter c, while are strongly dependent on the parameters x L and f. The new gauge osons, such as Z and B, can give corrections to R at tree-level and one-loop. The one-loop contriutions are smaller than the tree-level contriutions at least y two orders of magnitude in most of the parameter space. These contriutions are sensitive to the parameters c and f. Thus, the total correction of the LH model to the ranching ratio R is mainly dependent on the parameters f, c and x L. Thus, we can take the parameters c and ν ν c = 1 and ν ν = ν 5f for calculating the total correction to R. as fixed value: R LH R exp c'=0.1 c'=0.4 c'=0.707 c'= Figure 11: The predicted value of R LH parameter x L for four values of the mixing parameter c. x L in the LH model as a function of the mixing In Fig.10 we plot the ranching ratio R LH as a function of the mixing parameter c 18

19 for x L = 0.5 and four values of the scale parameter f. From Fig.10 we can see that the value of R LH decreases as the parameter c increasing. For f = 1TeV, the value of R is too large to consistent with the precision experimental value R exp in most of the parameter space. Furthermore, the large value of the scale parameter f is in favor of the general expectation ased on other phenomenological explorations. Thus, in Fig.11, we take f = 3TeV and plot the R LH as a function of the mixing parameter x L for four values of the parameter c. From Fig.11 we can see that the value of R LH the parameter x L increasing. If we demand that the predicted value R LH the precision experimental value R exp decreases as consistent with within σ ound for f = 3TeV, there must e: c = 0.1, 0.16 x L 0.67; c = 0.4, 0.5 x L 0.74; c = 1, 0.38 x L 0.84; c = 0.9, 0.39 x L If we take the small value for the scale parameter f, these constraints will ecame more strong. For example, for f = TeV and c = c = 1, we have 0.8 x L 0.65 in order to R LH consistent with R exp within σ ound. Little Higgs models have generated much interest as possile alternatives to weak scale supersymmetry. The LH model is a minimal model of this type, which realizes the little Higgs idea. In this paper, we study the corrections of the new particles predicted y the LH model to the ranching ratio R. We find that the corrections of the neutral scalars to R is very small, which can e neglected. The charged scalars can generated the negative corrections to R. The new gauge osons and fermions might generate the positive or negative corrections to R, which dependent on the values of the mixing parameters c, c and x L. If we demand that the contriutions of the new gauge osons and fermions cancel those generated y the charged scalars and make the predicted value R LH consistent with the precision experimental value R exp, then the parameters x L, c and f must e severe constrained. Acknowledgments 19

20 This work was supported in part y the National Natural Science Foundation of China ( ). 0

21 Appendix A: The masses of the gauge osons Z and B, triplet scalar Φ, and the vector-like quark T. The masses of the gauge osons Z, B and W can e written at the order of ν f : M Z = m Z C W [ f s c ν 1 5S3 W C W f MB = m ZSW[ 5s c ν 1 + 5C3 W 8S W M W = m Z c W ( f sc(c s + s c ) ], s c (5CW s c SW s c ) (38) s c (c s + s c ) ], sc(5cws c SWs c ) (39) 1), (40) s c ν where m Z is the mass of the SM gauge oson Z, ν is the electroweak scale. For the triplet scalar Φ, we have MΦ = m f 1 H ν 1 ( 4f ν (41) ν ), ν where m H is the SM Higgs mass and ν is the triplet scalar vacuum expectation value(vev). The mass of the heavy vector-like quark T can e written as: M T = m tf 1 ν [1 ν x L (1 x L ) f x L(1 + x L )], (4) where m t is the SM top quark mass, x L is the mixing parameter etween the SM top quark and the heavy vector-like quark T, which is defined as x L = λ 1. λ λ 1 and λ are 1 +λ the Yukawa coupling parameters. Appendix B: The relevant coupling constants of the gauge osons to fermions. Wt : g t L = g t R ie SW [1 ν f (x L + c (c s ))]V SM t, (43) = 0, (44) where V SM t is the SM CKM matrix element. In our calculation, we will take V SM t = 1. WT : W t : gl T = e ν SW f x LVt SM, gr T = 0. (45) gl t = e c SW s V t SM, gr t = 0. (46) 1

22 W T : L = e ν SW f x c L s V t SM, gr t = 0. (47) g t Z : gl = e { 1 S W C W S W + ν (c s ) f [c (c s )( c )]}, gr = e [ 1 S W C W 3 S W + 5 s )( 1 3 f (c 5 1 c )]. Ztt : gl t e = {1 S W C W 3 S W ν f [x L + c (c s ) 4 ν + 5 (c s )( 4 5 c + 3 s x L )]}, (48) (49) gr t = e { S W C W 3 S W ν s )[ 3 f 5(c 5 c (1 1 3 x L + 15 x L)]}. ZTT : gl T gr T e = ( S W C W 3 S W), (50) ZtT : L = i e x L ν S W C W 4f, g tt gtt R = 0. (51) ZW + W : g ZWW = g ZW W = ec W (5) S W Z : gl = e c S W s, g R = 0 (53) Z tt : g t L = e S W c s, gt R = 0. (54) Z TT : gl T = gt R = e ( S W C W 3 S W ). (55) B : g L = e 3C W s c (1 5 1 c ), g R = e B cc : g c L = e 3C W s c (1 5 1 c ), g c R = 3C W s c ( c ). (56) 4e 3C W s c (1 5 1 c ). (57) Appendix C: The coupling constants of the scalars to fermions. H 0 : Φ 0 : i m (1 4ν ν ν + ν f 3 ν f). (58) i m ( ν ν f 4ν ). (59) ν m Φ P : ( ν ν f 4ν ). ν (60) Φ + t : i [m t P L + m P R ]( ν ν f 4ν ), ν (61)

23 with P L = 1 γ 5, P R = 1+γ 5. Φ + T : i m t ( ν ν f 4ν ν ) xl P L. (6) 1 x L The coupling vertex of the SM gauge oson Z to the charged scalars Φ ± is e i S S W C W(P 1 P ) µ. W 3

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