Double Higgs production in the littlest Higgs Model with T-parity at high energy $e^{+}e^{-}$ Colliders

In the framework of the littlest Higgs model with T-parity(LHT), we investigate the double Higgs production processes $e^{+}e^{-}\rightarrow ZHH$ and $e^{+}e^{-}\rightarrow \nu\bar{\nu}HH$ at high energy $e^{+}e^{-}$ colliders. We calculate the production cross sections and find that the relative correction at the center-of-mass energy $\sqrt{s}=500$ GeV can maximally reach $-30%$ for the process $e^{+}e^{-}\rightarrow ZHH$ and $-16%$ for the process $e^{+}e^{-}\rightarrow \nu\bar{\nu}HH$ in the allowed parameter space, respectively. These large relative corrections can reach the detection range of the future $e^{+}e^{-}$ colliders so that they can be used to test the LHT effect. The two relevant decay modes $e^{+}e^{-}\rightarrow ZHH \rightarrow l\bar{l} b\bar{b}b\bar{b}$ and $e^{+}e^{-}\rightarrow \nu\bar{\nu}HH\rightarrow \nu\bar{\nu}b\bar{b}b\bar{b}$ are studied and some distributions of the signal and background are displayed.


INTRODUCTION
On the 4th of July 2012, ATLAS [1] and CMS [2] collaborations have announced the existence of a Higgs-like resonance around 125 GeV confirming the cornerstone of the Higgs mechanism[3] that predicted such particle long times ago. However, the discovery of a Higgs-like boson is not enough to fully understand the mechanism of electro-weak symmetry breaking (EWSB) and mass generation. The Higgs self-coupling is the key ingredient of the Higgs potential and its measurement is probably the most decisive test of the EWSB mechanism. To establish the Higgs mechanism unique experimentally, the Higgs potential of the Standard Model(SM) [4] must be reconstructed. In order to accomplish this, not only the Yukawa couplings and the Higgs-gauge couplings but also the Higgs self-couplings which include the trilinear coupling and the quartic coupling should be measured. The investigation of the Higgs self-couplings requires final states containing two or more Higgs bosons. In fact, the cross sections for three Higgs boson production processes are reduced by three order of magnitude compared to those for the double Higgs boson production [5] [6], the quartic Higgs self-coupling remains elusive. The phenomenology calculations show that it is difficult to measure the trilinear Higgs self-coupling at the Large Hadron Collider (LHC) due to the large QCD background [7]. But the e + e − linear colliders, such as the International Linear Collider (ILC) [8] and the Compact Linear Collider (CLIC) [9], have clean environment and can provide a possibly opportunity for studying the trilinear Higgs self-coupling [5].
The littlest Higgs model with T-parity(LHT) [10] was proposed as a possible solution to the hierarchy problem and so far remains a popular candidate of new physics. At the high energy e + e − colliders, there are two main processes for the SM Higgs boson, e + e − → ZHH and e + e − → ννHH, where the former reaches its cross-section maximum at a center-of-mass energy of around 500 GeV, while the cross-section for the latter is dominating above 1 TeV and increases towards higher energies. In the LHT model, some new particles are predicted and some couplings of the Higgs boson are modified.
These new effects will alter the property of the SM Higgs boson and influence various SM Higgs boson processes. The single Higgs production processes in the LHT model have been investigated in our previous work [11]. In this work, we will study the double Higgs production processes, e + e − → ZHH and e + e − → ννHH.
The paper is organized as follows. In Sec.II we give a brief review of the LHT model related to our work. In Sec.III we study the effects of the LHT model in the double Higgs boson productions and present some distributions of the final states. Finally, we give a short summary in Sec.IV.

II. A BRIEF REVIEW OF THE LHT MODEL
The LHT is a nonlinear σ model with a global symmetry under the SU(5) group and a gauged subgroup [SU(2) ⊗ U(1)] 2 . The SU(5) global symmetry is broken down to SO (5) by the vacuum expectation value (VEV) of the σ field, Σ 0 , given by After the global symmetry is broken, there arise 14 Goldstone bosons which are described by the "pion" matrix Π. The Goldstone bosons are then parametrized as where f is the breaking energy scale.
The σ field kinetic Lagrangian is given by [12] L with the [SU(2) ⊗ U(1)] 2 covariant derivative defined by where W µ j = 3 a=1 W µ a j Q a j and B µ j = B µ j Y j are the heavy SU(2) and U(1) gauge bosons, with Q a j and Y j the gauge generators, g j and g ′ j are the respective gauge couplings. In the gauge boson sector, T-parity is introduced as an exchange symmetry between the gauge bosons of the two different copies of the SM gauge group as The light(L) and heavy(H ) gauge fields can be obtained as The electroweak symmetry breaking SU(2) L × U(1) Y → U(1) em takes place via the usual Higgs mechanism. The mass eigenstates of the gauge fields are given by where θ W is the usual Weinberg angle and θ H is the mixing angle defined by At O(v 2 /f 2 ) in the expansion of the Lagrangian (3), the mass spectrum of the gauge bosons after EWSB is given by where and v SM = 246 GeV is the SM Higgs VEV. The global symmetries prevent the appearance of a potential for the scalar fields at tree level. The gauge and Yukawa interactions that break the global SO (5) symmetry induce radiatively a Coleman-Weinberg potential [14], V CW , whose explicit form can be obtained after expanding the Σ field where λ φ 2 , λ HφH and λ H 4 depend on the fundamental parameters of the model, whereas µ 2 , which receives logarithmic divergent contributions at one-loop level and quadratically divergent contributions at the two-loop level, is treated as a free parameter.
The HZZ, HW W and HHZZ, HHW W couplings involved in our calculations are modified at O(v 2 /f 2 ), which are given by
At the tree level, the Feynman diagrams relevant to the process e + e − → ZHH and the process e + e − → ννHH(ν = ν e , ν µ , ν τ ) are showed in Fig.1 and Fig.2, respectively. In both processes, we can see that only the first column of the diagrams, i.e. Fig.1(a   On the left panel of Fig.3, we show the dependance of the production cross sections σ of the processes e + e − → ZHH and e + e − → ννHH on the center-of-mass energy √ s for the scale f = 700 GeV in the LHT model and the SM, respectively. We can see that the e + e − → ZHH cross section decreases with increasing center-of-mass energy √ s while e + e − → ννHH cross section increases. The e + e − → ZHH cross section has the peak value around √ s ∼ 500 GeV. For √ s ∼ 1 TeV, the two cross sections are of the same order of magnitude, with e + e − → ννHH being the larger source of Higgs boson pairs for √ s ≥ 1 TeV. Since the ννHH production is peaked in the forward region, it is important to ensure that an efficient tagging of the HH → bbbb, W + W − W + W − decay can be achieved.
On the right panel of Fig.3, we show the dependance of the relative corrections δσ/σ of the processes e + e − → ZHH and e + e − → ννHH on the scale f for the center-of-mass energy √ s = 500 GeV. We can see that the relative correction δσ/σ of this two processes are both negative and decouple at the high scale f . Considering the lower bound on the scale f from the global fit of the latest experimental data [20], the relative correction δσ/σ of the process e + e − → ZHH can reach −30% ∼ −25% and the relative correction δσ/σ of the process e + e − → ννHH can reach −16% ∼ −12% for the scale f in the range 600GeV ∼ 700GeV. Furthermore, combined with the left panel of Fig.3, we can see that the relative correction δσ/σ of the process e + e − → ννHH increases with the center-ofmass energy √ s, which can reach over −30% for the center-of-mass energy √ s = 1500 GeV. These relative correction of the cross section are significant so that they may be observed at the future e + e − colliders with high integrated luminosity.
A. e + e − → ZHH → (ll)(bb)(bb) For light Higgs boson masses, the Higgs boson decays predominantly in a bb pair. The ZHH → qqbbbb final state benefits from a high statistics with ∼ 35% of the final states but requires a more complicated analysis. By contrast, though ZHH → llbbbb(l = e, µ) represents only ∼ 5% of the total final state, this topology produces an easy signature.
Therefore, we choose the llbbbb final state and display some normalized distributions in the LHT model. The experimental signature is very clean, namely four b-jets (two pairs with invariant Higgs mass) plus ll with invariant Z mass.
In Fig.4 we display the total transverse energy H T , the separation between the ll from Z boson (∆R ll ≡ (∆φ) 2 + (∆η) 2 ), the transverse momentums p b T and the invariant mass of four b-jets M HH in the LHT model. We can see that the peak of the total transverse energy is at H T ∼ 400 GeV and the peak of the ∆R ll is at ∆R ll ∼ 1.4. The p b1 T , p b2 T , p b3 T and p b4 T are arranged according to the size of the transverse momentum, where b1(b3) and b2(b4) incline to fly back-to-back. The M HH distribution is known to be sensitive to the Higgs boson self-coupling, in particular for small values of the Higgs-pair mass. Since it is impossible to know which b-jet has to be paired with whichb-jet when reconstructing the Higgs bosons in the event, here we give the four b-jets invariant mass distribution M HH .
B. e + e − → ννHH → νν(bb)(bb) Due to the dominant decay mode of Higgs is H → bb, the experimental signature for e + e − → ννHH is then four b-jets (two pairs with invariant Higgs mass) plus missing energy and momentum.
In Fig.5 we display the total transverse energy H T , the missing energy E T , the transverse momentums p b T and the invariant mass of four b-jets M HH in the LHT model. We can see that the peak of the total transverse energy is at H T ∼ 380 GeV and the peak of the missing energy is at E T ∼ 80 GeV. The behavior of the transverse momentums p b T for the four b-jets are similar to the process e + e − → ZHH → llbbbb. Likewise, we display the invariant mass distribution M HH of the four b-jets.

IV. SUMMARY
In this paper, we studied the double Higgs boson productions at high energy e + e − colliders in the LHT model. The two main production channels e + e − → ZHH and e + e − → ννHH have been investigated. For √ s = 500 GeV, we calculated the production cross section and found that the relative correction of the process e + e − → ZHH can reach −30% and the relative correction δσ/σ of the process e + e − → ννHH can reach −16% when the scale f is chosen as low as 600 GeV. This result may be a probe of the LHT model at the future high energy e + e − colliders. In order to investigate the observability, the decay modes e + e − → ZHH → llbbbb and e + e − → ννHH → ννbbbb were studied and some relevant final state distributions were presented.