New gauge bosons from the littlest Higgs model and the process e+e tt¯ − → Chong-Xing Yue, Lei Wang, Jian-Xing Chen Department of Physics, Liaoning Normal University, Dalian 116029. P.R.China ∗ 5 0 0 February 2, 2008 2 n a J 0 2 Abstract 1 v Inthecontext of thelittlest Higgs(LH) model, westudytheprocesse+e− → 6 8 tt¯. WefindthatthenewgaugebosonsZH andBH canproducesignificantcorrection 1 1 effects on this process, which can be further enhanced by the suitably polarized 0 5 beams. In most of the parameter space preferred by the electroweak precision data, 0 / the absolute value of the relative correction parameter R is larger than 5%. As h BH p long as 1TeV M 1.5TeV and 0.3 c 0.5, the absolute value of the - ≤ ZH ≤ ≤ ≤ p e relative correction parameter RZH is larger than 5%. With reasonable values of h : the parameters of the LH model, the possible signals of the new gauge bosons B v H Xi and ZH can be detected via the process e+e− tt¯in the future LC experiments → r a with the c.m. energy √S = 800GeV. BH exchange and ZH exchange can generate significantly corrections to the forward-backward asymmetry A (tt¯) only in small FB part of the parameter space. ∗ E-mail:[email protected] 1 I. Introduction Although the standard model(SM) that bases on thegaugegroup SU(2) U(1) has L Y × been successful in describing the physics of electroweak interactions, the mechanism of the electroweak symmetry breaking(EWSB) and the origins of the masses of the elementary fermions are still unknown. Furthermore, its scalar sector suffers from the problems of trivialityandunnaturalness, etc. Thus, itisquitepossiblethattheSM isonlyaneffective theory validbelow some high energy scale. New physics(NP) should exist at energy scales around TeV. Recently, a kind of theory for EWSB was proposed to solve the hierarchy between the TeV scale of possible NP and the electroweak scale v = 246GeV, which is known as ”little Higgs models”[1,2,3]. The key feature of these models is that the Higgs boson is a pseudo-Goldstone boson of a global symmetry which is spontaneously broken at some higher scale f and thus is naturally light. EWSB is induced by a Coleman-Weinberg potential, which is generated by integrating out the heavy degrees of freedom. This type of models can be regarded as one of the important candidates of the NP beyond the SM. A high energy e+e− linear collider(LC) will offer an opportunity to make precision measurement of the properties of the electroweak gauge bosons, top quarks, Higgs bosons andalsotoconstrainNP [4]. IntheLC experiments, topquarkpairsaremainlyproduced from the S-channel exchange of the SM gauge bosons γ and Z via the process e+e− tt¯ → [5]. The total cross section is of the order of 1pb, so that top quark pairs will be produced at large rates in a clean environment at LC. If we assume that the integrated luminosity £ is about 100fb−1, there will be several times 104 top quark pairs to be generated in int the future LC experiments. Furthermore, the QCD and EW corrections to the process e+e− tt¯are small and decrease as the centre-of-mass(c.m.) energy √S increasing. The → option of longitudinally polarized beams can help to improve the measurement precision and reduce background in search for NP. Thus, theoretical calculations of new particles contributions to the process e+e− tt¯are of much interest for testing of NP. → In general, the new gauge bosons are heavier than the current experimental limits on direct searches. However, thesenew particlesmay producevirtualeffects onsome physical 2 observable, which may be detected in the present or future high energy experiments. In Ref.[6], we discussed the possible of detecting the new gauge bosons Z and B predicted H H by the littlest Higgs(LH) model [1] in the future LC experiments with the c.m. energy √S = 500GeV and the integrating luminosity £ = 340fb−1 and both beams polarized int via considering their contributions to the processes e+e− ff¯ with f = τ,µ,b and → c. Since the masses of these fermions are largely smaller than the c.m. energy √S, we have neglected the masses of these fermions in our numerical estimations. Our results show that the new gauge bosons Z and B can indeed produce significant contributions H H to these process in most of the parameter space preferred by the electroweak precision data, which might be observable in the future LC experiments. The aim of this paper is to consider the contributions of the Z and B to the process e+e− tt¯and discuss H H → whether these new particles can be detected via this process in the future LC experiments with the c.m. energy √S = 800GeV and the integrating luminosity £ = 580fb−1. We int find that the absolute value of the relative correction parameter R generated by B BH H exchange is larger than 8% in most of the parameter space of the LH model preferred by the electroweak precision data. As long as 1TeV M 1.5TeV and 0.3 c 0.5, the ≤ ZH ≤ ≤ ≤ absolutevalueofR islargerthan5%. Ifweassumethattheinitialelectronandpositron ZH beams are suitably polarized, the absolute values of the relative correction parameters R and R can be enhanced. Thus, with reasonable values of the parameters of the BH ZH LH model, the possible signals of the new gauge bosons B and Z can be detected in H H the future LC experiments with the c.m. energy √S = 800GeV, which is similar to the conclusions given in Ref.[6]. We further calculate the contributions of these new gauge bosons to the forward-backward asymmetry A (tt¯). We find that they can generate FB significantly corrections to the forward-backward asymmetry A (tt¯) only in small part FB of the parameter space. In section II, we give the formula of the contributions of new gauge bosons B and Z H H to the process e+e− tt¯and estimate the values of the relative corrections parameters → R = σBH(tt¯)/σSM(tt¯) and R = σZH(tt¯)/σSM(tt¯). The dependence of the relative BH ZH correctionparametersR andR ontheinitialbeampolarizationisdiscussedinsection BH ZH 3 III.InsectionIV,we calculate thecontributions ofthese newgaugebosonstotheforward- backward asymmetry A (tt¯). Our conclusions and discussions are given in section V. FB II. Corrections of the new gauge bosons B and Z to the process e+e− tt¯ H H → The LH model [1] is one of the simplest and phenomenologically viable models, which realizes the little Higgs idea. It consists of a non-linear σ model with a global SU(5) symmetry, which is broken down to its subgroup SO(5) by a vacuum condensate f ∼ Λs/4π TeV. At the same time, the locally gauged group SU(2) U(1) SU(2) 1 1 2 ∼ × × × U(1) is broken to its diagonal subgroup SU(2) U(1), identified as the SM electroweak 2 × gauge group. This breaking scenario gives rise to four massive gauge bosons B , Z , and H H ± W , which might produce characteristic signatures at the present and future high energy H collider experiments [7,8,9]. Taking account of the gauge invariance of the Yukawa coupling and the U(1) anomaly cancellation, the coupling expressions of the gauge bosons B and Z to ordinary parti- H H cles, which are related to our calculation, can be written as [7]: 3e 2 e 2 gBHee = (c′2 ), gBHee = (c′2 ); (1) V 4C s′c′ − 5 A 4C s′c′ − 5 w w e 5 2 1 e 1 2 1 gBHtt = [ ( c′2) x ], gBHtt = [ ( c′2) x ]; (2) V 2C s′c′ 6 5 − − 5 L A 2C s′c′ 2 5 − − 5 L w w ec ec gZHee = , gZHee = ; (3) V −4S s A 4S s w w ec ec gZHtt = , gZHtt = . (4) V 4S s A −4S s w w WhereS = sinθ ,θ istheWeinbergangle. Usingthemixingparametersc(s = √1 c2) w w w − and c′(s′ = √1 c′2), we can represent the SM gauge coupling constants as g = g s = g c 1 2 − ′ ′ ′ ′ ′ and g = g s = g c. The mixing angle parameter between the SM top quark t and the 1 2 vector-like quark T is defined as x = λ2/(λ2 +λ2), in which λ and λ are the Yukawa L 1 1 2 1 2 coupling parameters. Global fits to the eletroweak precision data produce rather severe constraints on the parameter space of the LH model [10]. However, if the SM fermions are charged under U(1) U(1) , the constraints become relaxed. The scale parameter f = 1 2TeV is 1 2 × ∼ ′ allowed for the mixing parameters c, c, and x in the ranges of 0 0.5, 0.62 0.73, and L ∼ ∼ 4 0.3 0.6, respectively [11]. In this case, the masses of B and Z are allowed in the H H ∼ ranges of 300GeV 900GeV and 1TeV 3TeV, respectively. Thus, we will take the Z H ∼ ∼ ′ mass M , B mass M and the mixing parameters c, c and x as free parameters in ZH H BH L our calculation. 200 250 180 160 200 140 120 xL=0.3 xL=0.4 150 R(%)BH 10800 ccc’’’===000...667581 R(%)BH 100 60 40 50 20 0 -20 0 -40 -60 -50 400 500 600 700 800 900 400 500 600 700 800 900 MBH(GeV) MBH (GeV) (a) (b) 300 250 250 200 200 xL=0.5 xL=0.6 %) %) 150 R(BH 150 R(BH 100 100 50 50 0 0 -50 -50 400 500 600 700 800 900 400 500 600 700 800 900 MBH(GeV) MBH(GeV) (d) (c) Figure 1: The relative correction parameter R as a function of the B mass M for BH H BH ′ different values of the mixing parameters c and x . L For the SM, top quark pair tt¯ can be produced in sufficient abundance in the LC experiments. The main production mechanism proceed at the Born level by the S-channel annihilation of an initial electron-position pair into virtual photon or neutral gauge boson 5 Z, and their subsequent splitting into top quark pairs, e+e− γ,Z tt¯. For the LH → → model, the B exchange and Z exchange can also produce the top quark pairs. The H H production cross sections can be written as: Nfβ β2 4 S(M2 S) σBH(tt¯) = c (1 ) e2gBHeegBHtt BH − 8πS { − 3 3 V V (S M2 )2 +M2 Γ2 − BH BH BH β2 +[(gBHee)2 +(gBHee)2][(1 )[(gBHtt)2 +(gBHtt)2] (1 β2)(gBHtt)2] V A − 3 V A − − A S2 +(gZeegBHee +gZeegBHee) (S M2 )2 +M2 Γ2 V V A A − BH BH BH β2 [(1 )(gZttgBHtt +gZttgBHtt) (1 β2)(gBHtt)(gZtt)] − 3 V V A A − − A A 2S2[(S M2)(S M2 )+M Γ M Γ ] − Z − BH Z Z BH BH , (5) [(S M2)2 +M2Γ2][(S M2 )2 +M2 Γ2 ]} − Z Z Z − BH BH BH Nfβ β2 4 S(M2 S) σZH(tt¯) = c (1 ) e2gZHeegZHtt ZH − 8πS { − 3 3 V V (S M2 )2 +M2 Γ2 − ZH ZH ZH β2 +[(gZHee)2 +(gZHee)2][(1 )[(gZHtt)2 +(gZHtt)2] (1 β2)(gZHtt)2] V A − 3 V A − − A S2 +(gZeegZHee +gZeegZHee) (S M2 )2 +M2 Γ2 V V A A − ZH ZH ZH β2 [(1 )(gZttgZHtt +gZttgZHtt) (1 β2)(gZHtt)(gZtt)] − 3 V V A A − − A A 2S2[(S M2)(S M2 )+M Γ M Γ ] − Z − ZH Z Z ZH ZH (6) [(S M2)2 +M2Γ2][(S M2 )2 +M2 Γ2 ]} − Z Z Z − ZH ZH ZH with e e gZee = ( 1+4S2), gZee = (7) V 4S C − w A 4S C w w w w e 8 e gZtt = (1 S2), gZtt = , (8) V 4S C − 3 w A 4S C w w w w where β = 1 4m2t, m is the top quark mass. Γ represent the total decay widths of q − S t i the gauge bosons Z,Z , and B . Γ and Γ have been given in Ref.[6]. From above H H ZH BH equations, we cansee thatσBH(tt¯)mainlydependents thefreeparameters M , c′ andx , BH L while σZH(tt¯) only dependents the free parameters c and M , which is differently from ZH those for the process e+e− ff¯with f = τ,µ,b and c. In that case, the contributions → of the gauge bosons B is independent of the mixing parameter x . Thus, in this paper, H L ′ we will take the mixing parameters c,c and x as free parameters. Certainly, due to L 6 the mixing between the gauge bosons Z and Z , the SM tree-level couplings Zee¯ and H Ztt¯ receive corrections at the order of v2/f2, which can also produce contributions to the production cross section of the process e+e− tt¯. However, the contributions are → suppressed by the factor v4/f4, which are smaller than those of B or Z . Thus, we have H H neglected this kind of corrections in above equations. 40 c=0.10 30 c=0.30 %) c=0.50 ( H Z R 20 - 10 0 1000 1500 2000 2500 3000 M (GeV) ZH Figure 2: The relative correction parameter R as a function of the Z mass M for ZH H ZH three values of the mixing parameter c. To see the correction effects of B exchange and Z exchange on the tt¯production H H cross section, we plot the relative correction parameters R = σBH(tt¯)/σSM(tt¯) and BH R = σZH(tt¯)/σSM(tt¯) as functions of M andM in Fig.1 and Fig.2, respectively. ZH BH ZH From these figures, we can see that the gauge boson Z decreases the SM tt¯production H crosssectionσSM(tt¯)inalloftheparameterspace,whichsatisfiestheelectroweakprecision constraints. In most part of the parameter space, the absolute value of the relative correction parameter R is smaller than 5%, which is very difficult to be detected in the ZH 7 future LC experiments. This is consistent with the contributions of Z to the process H e+e− ff¯, which has been studied in Ref.[6]. However, for the gauge boson B , it H → is not this case. For M 800GeV, B exchange produce positive corrections to the BH ≤ H tt¯ production cross section σSM(tt¯) and the value of R increase as M , x and c′ BH BH L increasing. For 800GeV < M 900GeV, B exchange decrease the cross section BH ≤ H σSM(tt¯) and the absolute of R increase as M decreasing and x , c′ increasing. The BH BH L peak of the R resonance emerges when the B mass M is approximately equal to BH H BH the c.m. energy √S = 800GeV. In most part of the parameter space, the absolute value of R is larger than 8%. Thus, the virtual effects of B on the process e+e− tt¯should BH H → be easy detected in the future LC experiment with √S = 800GeV and £ = 580fb−1. int III. The dependence of the relative correction parameters R and R on the BH ZH electron and positron beam polarization An LC has a large potential of the discovery of new particles and is well suited for the precise analysis of NP beyond the SM. At present, the existing proposals are designed with high luminosity of about £ = 340fb−1 at √S = 500GeV and £ = 580fb−1 at int int √S = 800GeV [4]. An important tool of an LC is the use of polarized beams. Beam polarization is not only useful for a possible reduction of the background, but might also serve as a possible tool to disentangle different contributions to the signal and lead to substantial enhancement of the produce cross sections of some processes [12]. To see whether the contributions of the new gauge bosons B and Z to the process e+e− tt¯ H H → can indeed be detected, we discuss the dependence of the relative correction parameters R and R on the initial electron and positron beam polarization in this section. BH ZH Considering the polarization of the initial electron and positron beams, the cross sec- tion of the process e+e− tt¯can be generally written as: → σ(tt¯) = (1+P )(1 P )(σ (tt¯)+σ (tt¯))+(1 P )(1+P )(σ (tt¯)+σ (tt¯)), (9) e e¯ RR RL e e¯ LL LR − − where P and P are the degrees of longitudinal electron and position polarization, respec- e e¯ tively. σ are the chiral cross sections of this process. The relative correction parameters ij ′ R andR areplottedasfunctionsofM andM forc = 0.65,x = 0.5,c = 0.3and BH ZH BH ZH L 8 different beam polarizations in Fig.3 and Fig.4, respectively. In these two figures, we have used the solid line, dashed line, and dotted line to represent (P ,P )=(0,0), (0.8, 0.6), e e¯ − and ( 0.8,0.6), respectively. Our calculation results show that the absolute values of − R [R ] for (P ,P ) = (0.8,0.6)[(-0.8,-0.6)] are smaller than those for (P ,P ) = (0,0). BH ZH e e¯ e e¯ Thus, in Fig.3 and Fig.4 we do not plot these lines. 80 60 xL=0.5 40 ( 0 , 0 ) (0.8,-0.6) ) 20 (-0.8,0.6) % ( H B 0 R -20 -40 -60 -80 -100 400 500 600 700 800 900 M (GeV) BH Figure 3: The relative correction parameter R as a function of the B mass M for BH H BH ′ c = 0.65, x = 0.5, and (P ,P ) = (0,0),(0.8, 0.6),( 0.8,0.6). L e e¯ − − From Fig.3 and Fig.4 we can see that the suitably polarized beams can indeed enhance the virtual effects of the new gauge bosons B and Z on the process e+e− tt¯. In the H H → whole parameter space preferred by the electroweak precision data, the value of R for BH (P ,P ) = (0.8, 0.6) is larger than that for (P ,P ) = (0,0), while the absolute values of e e¯ e e¯ − R for (P ,P ) = ( 0.8,0.6) is larger than that for (P ,P ) = (0,0). Varying the values ZH e e¯ − e e¯ ′ of the free parameters c, x , and c does not change this conclusion. So, in Fig.3 and L ′ Fig.4 we have taken these parameters for fixed values x = 0.5, c = 0.65, and c = 0.3. L Certainly, the values of R and R change as the values of these parameters varying. BH ZH 9 For example, for 0.3 c 0.5 and 1TeV M 2TeV, the absolute value of R for ≤ ≤ ≤ ZH ≤ ZH (P ,P ) = ( 0.8,0.6) is larger than 6%. The absolute of R for (P ,P ) = (0.8, 0.6) is e e¯ − BH e e¯ − ′ larger than 5% for x = 0.5, 0.68 c 0.73 and 500GeV < M 900GeV, but for L ≤ ≤ BH ≤ ′ x = 0.6 its value is larger than 5% for 0.65 c 0.73 and 450GeV M 900GeV. L ≤ ≤ ≤ BH ≤ Thus, using the suitably polarization of the initial electron and positron beams, it is more easy to detect the possible signals of the new gauge bosons B and Z in the future LC H H experiments. 50 40 30 ) % ( 0 , 0 ) ( H (0.8,-0.6) Z R (-0.8,0.6) - 20 10 0 1000 1500 2000 2500 3000 M (GeV) ZH Figure 4: The relative correction parameter R as a function of the Z mass M for ZH H ZH c = 0.3 and (P ,P ) = (0,0),(0.8, 0.6),( 0.8,0.6). e e¯ − − IV. Gauge bosons B , Z and the forward-backward asymmetry A (tt¯) H H FB The events generated by the process e+e− ff¯can be characterized by the momen- → tum direction of the emitted fermion. If we assume that the final state fermion travels forward(F) or backward(B) with respect to the electron beam, thanthe forward-backward asymmetry can be defined as: σ σ F B A = − , (10) FB σ +σ F B 10