In the framework of the littlest Higgs Model with T-parity, we discuss the top partner production at future e+e- collider. We calculate the cross sections of the top partner production processes and associated production processes of Higgs and top partner under current constraints. Then, we investigate the observability of the T-odd top partner pair production through the process e+e-→T-T¯-→tt¯AHAH in the tt¯ dilepton channel for two T-odd top partner masses mT-=603(708) GeV at s=1.5 TeV. We analyze the signal significance depending on the integrated luminosity and find that this signal is promising at the future high energy e+e- collider.
National Natural Science Foundation of China1140504711404099Henan Normal Universityqd152071. Introduction
The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2] is a great step towards understanding the electroweak symmetry breaking (EWSB) mechanism. However, the little hierarchy problem [3, 4], which is essentially from quadratically divergent corrections to the Higgs mass parameter, still exists. In the past, various new physics models have been proposed to solve this problem, and the littlest Higgs Model with T-parity (LHT) [5–7] is one of the most promising candidates.
In the LHT model, the Higgs boson is constructed as a pseudo-Nambu-Goldstone particle of the broken global symmetry. The quadratic divergence contributions to Higgs boson mass from the SM top quark loop, gauge boson loops, and the Higgs self-energy are cancelled by the corresponding T-parity partners, respectively. Among the partners, the top partner is the most important one since it is responsible for cancelling the largest quadratically divergent correction to the Higgs mass induced by the top quark.
Recently, the ATLAS and CMS collaborations have performed the searches for the vector-like top partner through the pair or single production with three final states bW, tZ, and tH and have excluded the top partner with the mass less than about 700 GeV [8–10]. Besides, a search has been performed in pair-produced exotic top partners, each decay to an on-shell top (or antitop) quark and a long-lived undetected neutral particle [11]. Apart from direct searches, the indirect searches for the top partners through their contributions to the electroweak precision observables (EWPOs) [12, 13], Z-pole observables [14–16], and the flavor physics [17–24] have been extensively investigated. The null results of the top partners, in conjunction with the EWPOs and the recent Higgs data, have tightly constrained the parameter space of the LHT model [25–30].
Compared to the hadron colliders, e+e- linear colliders may provide cleaner environments to study productions and decays of various particles. Some design schemes have been put forward, such as the International Linear Collider (ILC) [31–33] and the Compact Linear Collider (CLIC) [34–36]; they can run at the center of mass (c.m.) energy ranging from 500 GeV to 3000 GeV, which enables us to perform precision measurements of the top partner above the threshold. In addition, the polarization of the initial beams at e+e- linear colliders will be useful to study the properties of the top partner. Some relevant works have been widely studied in various extensions of the Standard Model (SM) [37–39], including the Little Higgs model [40, 41]. However, the works in Little Higgs model mostly were performed many years ago and before the discovery of the Higgs boson, so it is necessary to revisit this topic. Moreover, the different final states are analyzed in this work.
The paper is organized as follows. In Section 2 we review the top partner in the LHT model. In Section 3 we calculate top partner production cross sections. In Section 4 we investigate signal and discovery potentiality of the top partner production at e+e- collider. Finally, we draw our conclusions in Section 5.
2. Top Partner in the LHT Model
The LHT model is a nonlinear σ model based on the coset space SU(5)/SO(5) [42–49]. The global group SU(5) is spontaneously broken into SO(5) at the scale f~O (TeV) by the vacuum expectation value (VEV) of the Σ field, which is given by(1)Σ0=Σ02×2012×201012×2002×2.The VEV Σ0 also breaks the gauged subgroup SU(2)×U(1)2 of SU(5) down to the diagonal SM electroweak symmetry SU(2)L×U(1)Y. After the symmetry breaking, there arise 4 new heavy gauge bosons WH±,ZH,AH whose masses are given at O(v2/f2) by(2)MWH=MZH=gf1-v28f2,MAH=g′f51-5v28f2 with g and g′ being the SM SU(2)L and U(1)Y gauge couplings, respectively. The heavy photon AH is the lightest T-odd particle and can serve as a candidate for dark matter. In order to match the SM prediction for the gauge boson masses, the VEV v needs to be redefined as(3)v=f2arccos1-vSM2f2≃vSM1+112vSM2f2, where vSM = 246 GeV.
In the fermion sector, the implementation of T-parity requires the existence of mirror partners for each original fermion. In order to do this, two fermion SU(2) doublets q1 and q2 are introduced and T-parity interchanges these two doublets. A T-even combination of these doublets is taken as the SM fermion doublet and the T-odd combination is its T-parity partner. The doublets q1 and q2 are embedded into incomplete SU(5) multiplets Ψ1 and Ψ2 as Ψ1=(q1,0,02)T and 2=(02,0,q2)T, where 02=(0,0)T. To give the additional fermions masses, an SO(5) multiplet Ψc is also introduced as Ψc=(qc,χc,q~c)T, whose transformation under the SU(5) is nonlinear: Ψc→UΨc, where U is the unbroken SO(5) rotation in a nonlinear representation of the SU(5). The components of the latter Ψc multiplet are the so-called mirror fermions. Then, one can write down the following Yukawa-type interaction to give masses of the mirror fermions:(4)Lmirror=-κijfΨ¯2iξ+Ψ¯1iΣ0Ωξ†ΩΨcj+h.c., where i,j=1,2,3 are the generation indices. The masses of the mirror quarks uHi,dHi and mirror leptons lHi,νHi up to O(v2/f2) are given by(5)mdHi=2κif,muHi=mdHi1-v28f2,mlHi=2κif,mνHi=mlHi1-v28f2, where κi are the diagonalized Yukawa couplings.
In the top quark sector, two singlet fields TL1 and TL2 (and their right-handed counterparts) are introduced to cancel the large radiative correction to the Higgs mass induced by the top quark. Both fields are embedded together with the q1 and q2 doublets into the SU(5) multiplets: Ψ1,t=(q1,TL1,02)T and Ψ2,t=02,TL2,q2T. The T-even combination of qi is the SM fermion doublet and the other T-odd combination is its T-parity partner. Then, the T-parity invariant Yukawa Lagrangian for the top sector can be written down as follows:(6)Lt=-λ1f22ϵijkϵxyΨ¯1,tiΣjxΣky-Ψ¯2,tΣ0iΣjx′Σky′tR′-λ2fT¯L1TR1+T¯L2TR2+h.c.,where ϵijk and ϵxy are the antisymmetric tensors with i,j,k=1,2,3 and x,y=4,5, Σ′=ΣΩΣ†ΩΣ is the image of Σ under T-parity, and λ1 and λ2 are two dimensionless top quark Yukawa couplings. Under T-parity, these fields transform as TL1↔-TL2, TR1↔-TR2, and tR′→tR′. The above Lagrangian contains the following mass terms:(7)Lt⊃-λ1fsΣ2t¯L+tR′+1+cΣ2T¯L+′tR′-λ2fT¯L+′TR+′+T¯L-′TR-′+h.c., where cΣ=cos(2h/f) and sΣ=sin(2h/f). The T-parity eigenstates have been defined as tL+=(tL1-tR1)/2, TL±′=(TL1∓TL2)/2, and TR±′=(TR1∓TR2)/2. Note that T-odd Dirac fermion T-≡(TL-′,TR-′) does not have the tree-level Higgs boson interaction, and thus it does not contribute to the Higgs mass at one-loop level.
The two T-even eigenstates (tL+,tR′) and (TL+′,TR+′) mix with each other so that the mass eigenstates can be defined as(8)tL=cosβtL+-sinβTL+′,TL+=sinβtL++cosβTL+′,tR=cosαtR′-sinαTR+′,TR+=sinαtR′+cosαTR+′,where the mixing angles α and β can be defined by the dimensionless ratio R=λ1/λ2 as(9)sinα=R1+R2,sinβ=R21+R2vf. The t≡(tL,tR) quark is identified with the SM top quark, and T+≡(TL+,TR+) is its T-even heavy partner, which is responsible for the cancellation of the quadratic divergence to the Higgs mass induced by the top quark loop.
The Yukawa term generates the masses of the top quark and its partners, which are given at O(v2/f2) by(10)mt=λ2vR1+R21+v2f2-13+12R21+R22,mT+=fvmt1+R2R1+v2f213-R21+R22,mT-=fvmt1+R2R1+v2f213-12R21+R22.Since the T+ mass is always larger than the T-odd top partner T- mass, the T+ can decay into AHT- in addition to the conventional decay modes (Wb,tZ,tH).
The T-invariant Lagrangians of the Yukawa interactions of the down-type quarks and charged leptons can be constructed by two possible ways, which are denoted as Case A and Case B, respectively [50]. In the two cases, the corrections to the Higgs couplings with the down-type quarks and charged leptons with respect to their SM values are given at order OvSM4/f4 by (d≡d,s,b,li±)(11)ghd¯dghd¯dSM=1-14vSM2f2+732vSM4f4Case A,ghd¯dghd¯dSM=1-54vSM2f2-1732vSM4f4Case B.
3. Top Partner Production in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M134"><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo mathvariant="bold">+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo mathvariant="bold">-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> Collision
In the LHT model, the Feynman diagrams of top partner production are shown in Figure 1, which proceeds through the s-channel γ and Z exchange diagrams. These processes include T-even top partner pair production e+e-→T+T¯+, T-odd top partner pair production e+e-→T-T¯-, and a T-even top partner associating with a top quark production e+e-→tT¯+.
Feynman diagrams of the top partner production at e+e- collider.
The Feynman diagrams of the Higgs and top partner associated production are shown in Figure 2, which has additional diagrams mediated by the T-even top partner T+ compared to the process e+e-→tt¯H in the SM. These processes include Higgs associating with T-even top partner pair production e+e-→T+T¯+H, Higgs associating with T-odd top partner pair production e+e-→T-T¯-H, and Higgs associating with a top quark and a T-even top partner production e+e-→tT¯+H.
Feynman diagrams of the Higgs and top partner associated production at e+e- collider.
Before calculating the top partner production cross section, we firstly consider the constraints on the top partner mass from current measurements. We update the constraint on the LHT parameter in our previous works [51, 52], where the global fit of the latest Higgs data, EWPOs, and Rb measurements is performed. Thereinto, the constraints from the direct searches for Higgs data at Tevatron [53, 54] and LHC [55, 56] are obtained by the package HiggsSignals-1.4.0 [57, 58], which is linked to the HiggsBounds-4.2.1 [59–63] library. We compute the χ2 values by the method introduced in [64–66] and obtained the constraint on the LHT parameter space. This constraint will lead to the exclusion limits on the top partner masses, which is displayed on the R~f plane for Case A and Case B in Figure 3 at 2σ confidence level with δχ2=8.02. We can see that the combined constraints can, respectively, exclude mT+ and mT- up to (12)mT+>920750GeVCase A (B),mT->590500GeVCase A (B).One can notice that Case B predicts a stronger suppression for the down-type fermion couplings to the Higgs boson, such as Hbb¯, which helps to enhance the branching ratios of H→γγ,WW∗,ZZ∗,ττ, so that Case B is favored by the experimental data [67].
Exclusion limits on the top partner masses on the R~f plane at 2σ confidence level for Case A and Case B, where the parameter κ is marginalized over.
In the left frame of Figure 4, we show the top partner production cross sections as a function of c.m. energy s for f=700 GeV and R=1 (corresponding to mT+=986 GeV and mT-=708 GeV) in e+e- collision with unpolarized beams. The production cross sections are calculated at tree-level by using CalcHEP 3.6.25 [68, 69], where the SM parameters are taken as follows [70]:(13)sin2θW=0.231,αe=1128,MZ=91.1876GeV,mt=173.5GeV,mH=125GeV. We can see that the top partner pair production cross sections increase abruptly at threshold and reach a maximum roughly 200 GeV above threshold. Then, the production cross sections fall roughly with the c.m. energy s increase due to the s-channel suppression. The T-T¯- production usually has a larger cross section than T+T¯+ production since the T- mass is always lighter than the T+ mass in the LHT model. The production cross sections of the associated production of Higgs and top partner have the similar behavior as the top partner pair production, but usually have smaller cross sections due to smaller phase space. The production cross section of the process e+e-→tT¯+H reaches its maximum when the resonance decay of the top partner T+ emerges.
Top partner production cross sections as a function of s for f=700 GeV, R=1 in e+e- collision with (un)polarized beam.
Considering the polarization of the initial electron and positron beams, the cross section at e+e- collider can be expressed as [71](14)σ=141+pe1+pe¯σRR+1-pe1-pe¯σLL+1+pe1-pe¯σRL+1-pe1+pe¯σLR, where σRL is the cross section for completely right-handed polarized e- beam (pe=+1) and completely left-handed polarized e+ beam (pe¯=-1), and other cross sections σRR, σLL, and σLR are defined analogously. We show the top partner production cross sections in polarized beam with pe=0.8 and pe¯=-0.6 in the right frame of Figure 4 and find that the relevant top partner production cross sections can be enhanced by the polarized beams.
4. Signal and Discovery Potentiality
Take into account the relatively large production cross section; we will perform the Monte Carlo simulation and explore the sensitivity of T-odd top partner production in the following section. The T-odd top partner T- has a simple decay pattern, which decays almost 100% into the AHt mode. We will explore the sensitivity of T-odd top partner pair production with unpolarized beam through the channel(15)e+e-⟶T-T¯-⟶t⟶l+νlbt¯⟶l-νl¯b¯AHAH⟶l+l-+2b+ET which implies that the events contain one pair of oppositely charged leptons l+l-(l=e,μ) with high transverse momentum, two high transverse momentum b-jets, and large missing transverse energy ET.
The dominant background arises from e+e-→tt¯ in the SM. Besides, the most relevant backgrounds come from tt¯Z(→νν¯), W+(→l+νl)W-(→l-νl¯)Z(→bb¯), and W+(→l+νl)W-(→l-νl¯)H(→bb¯). Here, the backgrounds ZZZ, ZZH, and ZHH are neglected due to their small cross sections. We turn off the parton-level cuts and generate the signal and background events by using MadGraph 5 [72], where the UFO [73] format of the LHT model has been obtained by FeynRules [74] in [25]. We use MadGraph 5 to generate the process by issuing the following commands:
The parton shower and hadronization are performed with PYTHIA [75], and the fast detector simulations are performed with Delphes [76]. We use the default card (i.e., delphes_card_ILD) of ILC in Delphes 3.3.3. The b-jet tagging efficiency is taken as default value in delphes, where it is parameterized as a function of the transverse momentum and rapidity of the jets. When generating the parton-level events, we assume μR=μF to be the default event-by-event value. FastJet [77] is used to define jets via the anti-kt algorithm [78] with distance parameter ΔR=0.4. We use MadAnalysis 5 [79] for analysis, where the (mis)tagging efficiencies and fake rates are assumed to be their default values.
Take into consideration the constraints on the top partner mass from current measurements; we take f = 700 GeV, R=1 (corresponding to mT-=708 GeV) and f=700 GeV, R=1.5 (corresponding to mT-=603 GeV) for two benchmark points in the following calculations. In order to reduce the background contribution and enhance the signal contribution, some cuts of kinematic distributions are needed. In Figure 5, we show the normalized distributions of transverse momentum pTl1, the pseudorapidity ηl1, ηb1, the separation ΔR(l1,b1) between l1 and b1, the energy E(b1l1)(=E(b1)+E(l1)), and the total transverse energy HT.
Normalized distributions of ηl1, ηb1, pTl1, ΔRl1b1, E(b1l1), and HT in the signal and backgrounds for the two signal benchmark points at s=1.5 TeV.
Since the dominant background arises from tt¯, the cuts that are chosen to suppress the backgrounds should be centered around the tt¯ background. Firstly, we can apply the cuts of general kinematic distributions, such as pTl1, ηl1, and ηb1, to suppress the backgrounds. For the ΔR(l1,b1) distribution, there are two peaks in the tt¯, tt¯Z backgrounds and one peak in the WWZ, WWH backgrounds; we can use the deviation between the signal peak and background peak to suppress the backgrounds. Then, in view of the energy E(b1l1) distribution, we can also use the deviation between the signal peak and background peak to reduce the backgrounds. After that, the HT distribution of the signal can be utilized to remove the tt¯ background effectively. According to the above analysis, events are selected to satisfy the following cuts:(16)Cut-1: pTl1>50GeV;Cut-2: ηl1<1;ηb1<1;Cut-3: ΔRl1,b1<2.5;Cut-4: Eb1l1<400GeV;Cut-5:HT<400GeV;
For easy reading, we summarize the cut-flow cross sections of the signal and backgrounds for c.m. energy s = 1.5 TeV in Table 1. To estimate the observability quantitatively, the Statistical Significance (SS) is calculated after final cut by using Poisson formula [80](17)SS=2LS+Bln1+SB-S, where S and B are the signal and background cross sections and L is the integrated luminosity. The results for the SS values depending on the integrated luminosity for s = 1.5 TeV are shown in Figure 6. It is clear from Figure 6 that we can obtain the 2σ significance at a luminosity of 35(45) fb-1, 3σ significance at a luminosity of 70(100) fb-1, and 5σ significance at a luminosity of 200(280) fb-1 for mT- = 603(708) GeV.
Cut flow of the cross sections for the signal (S) and the backgrounds (B) for the two signal benchmark points (P1: f=700 GeV, R=1) and (P2: f=700 GeV, R=1.5) at s=1.5 TeV.
Cuts
S (×10-3 fb)
B (×10-3 fb)
S/B
T-T-- (P1)
T-T-- (P2)
tt-
tt-Z
WWZ
WWH
P1
P2
No cut
184
119
3485
32
367
100
0.046
0.03
Cut-1
139.8
94.0
2011
20.8
283
104
0.058
0.039
Cut-2
81.1
54.9
929.6
9.6
59.8
41.1
0.078
0.053
Cut-3
62.4
45.6
334.7
5.6
15.6
11.5
0.17
0.12
Cut-4
48.7
36.5
120.1
3.4
3.0
2.2
0.38
0.28
Cut-5
44.8
33.6
34.8
2.4
2.6
1.5
1.08
0.81
The statistical significance depending on integrated luminosity for s = 1.5 TeV.
5. Conclusions
In this paper, we discuss the top partner production at future e+e- collider in the LHT model. We first consider the constraints on the top partner masses from the current measurements and then calculate the cross sections of various top partner production processes, which include e+e-→T+T¯+, e+e-→T-T¯-, e+e-→tT¯+ and e+e-→T+T¯+H, e+e-→T-T¯-H, and e+e-→tT¯+H. Next, we investigate the observability of the T-odd top partner pair production through the process e+e-→T-T¯-→tt¯AHAH with the dilepton decay of the top quark pair for s = 1.5 TeV. We display the signal significance depending on the integrated luminosity and find that the 2σ significance can be obtained at a luminosity of 70(100) fb-1 for mT- = 603(708) GeV, which is promising at the future high energy e+e- collider with high luminosity.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (NNSFC) under Grants no. 11405047 and no. 11404099 and by the Startup Foundation for Doctors of Henan Normal University under Grant no. qd15207.
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