1 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 [3–6] that predicted such particle long times ago. However, the discovery of a Higgs-like boson is not enough to fully understand the mechanism of electroweak 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 uniquely establish the Higgs mechanism experimentally, the Higgs potential of the Standard Model (SM) [7–9] 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 orders of magnitude compared to those for the double Higgs boson production [10–18], 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 [19]. But the \(e^{+}e^{-}\) linear colliders, such as the international linear collider (ILC) [20] and the compact linear collider (CLIC) [21], have a clean environment and provide a possible opportunity for studying the trilinear Higgs self-coupling [10–14].

The littlest Higgs model with T-parity (LHT) [22–25] 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, \(e^{+}e^{-}\rightarrow ZHH\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\), 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, where the double Higgs production processes can provide a good opportunity to discriminate between the product group and simple group little Higgs models [26]. The single Higgs production processes in the LHT model have been investigated in our previous work [27]. In this work, we will study the double Higgs production processes, \(e^{+}e^{-}\rightarrow ZHH\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\).

The paper is organized as follows. In Sect. 2 we give a brief review of the LHT model related to our work. In Sect. 3 we study the effects of the LHT model in the double Higgs boson productions and present some distributions of the signal and background. Finally, we give a short summary in Sect. 4.

2 A brief review of the LHT model

The LHT is a nonlinear \(\sigma \) model with a global symmetry under the \(SU(5)\) group and a gauged subgroup \([SU(2) \otimes U(1)]^2\). The \(SU(5)\) global symmetry is broken down to \(SO(5)\) by the vacuum expectation value (VEV) of the \(\sigma \) field, \(\Sigma _0\), given by

$$\begin{aligned} \Sigma _0=\langle \Sigma \rangle \begin{pmatrix} \mathbf{0}_{2\times 2} &{}\quad 0 &{}\quad \mathbf{1}_{2\times 2} \\ 0 &{}\quad 1 &{}\quad 0 \\ \mathbf{1}_{2\times 2} &{}\quad 0 &{}\quad \mathbf{0}_{2\times 2} \end{pmatrix}. \end{aligned}$$
(1)

After the global symmetry is broken, there arise 14 Goldstone bosons, which are described by the “pion” matrix \(\Pi \). The Goldstone bosons are then parameterized as

$$\begin{aligned} \Sigma = \mathrm{e}^{i\Pi /f}\ \Sigma _0\ \mathrm{e}^{i\Pi ^T/f}\equiv \mathrm{e}^{2i\Pi /f}\ \Sigma _0, \end{aligned}$$
(2)

where \(f\) is the breaking energy scale.

The \(\sigma \) field kinetic Lagrangian is given by [28]

$$\begin{aligned} \mathcal {L}_\mathrm{K}= \frac{f^{2}}{8} \mathrm{Tr} | D_{\mu } \Sigma |^2, \end{aligned}$$
(3)

with the \([SU(2)\otimes U(1)]^2\) covariant derivative defined by

$$\begin{aligned}&D_{\mu } \Sigma = \partial _{\mu } \Sigma \nonumber \\&\quad - i \sum _{j=1}^2[ g_{j} W_{j\,\mu }^{a} (Q_{j}^{a}\Sigma + \Sigma Q_{j}^{a\,T}) + g'_{j} B_{j\,\mu } (Y_{j} \Sigma \!+\!\Sigma Y_{j}^{T})],\nonumber \\ \end{aligned}$$
(4)

where \(W_{j}^\mu = \sum _{a=1}^{3} W_{j}^{\mu \, a} Q_{j}^{a}\) and \(B_{j}^\mu = B_{j}^{\mu } Y_{j}\) are the heavy \(SU(2)\) and \(U(1)\) gauge bosons, with \(Q_j^a\) 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

$$\begin{aligned} W_{1\mu }^a \longleftrightarrow W_{2\mu }^a,\quad B_{1\mu } \longleftrightarrow B_{2\mu }. \end{aligned}$$
(5)

The light (L) and heavy (H) gauge fields can be obtained as

$$\begin{aligned} W_\mathrm{L}^a&= \frac{W_1^a + W_2^a}{\sqrt{2}},\quad B_\mathrm{L} = \frac{B_1 + B_2}{\sqrt{2}}, \nonumber \\ W_\mathrm{H}^a&= \frac{W_1^a - W_2^a}{\sqrt{2}},\quad B_\mathrm{H} = \frac{B_1 - B_2}{\sqrt{2}}. \end{aligned}$$
(6)

The electroweak symmetry breaking \(SU(2)_L \times U(1)_Y \rightarrow U(1)_{em}\) takes place via the usual Higgs mechanism. The mass eigenstates of the gauge fields are given by

$$\begin{aligned}&\!\!\!W_\mathrm{L}^{\pm } = \frac{W_\mathrm{L}^1 \mp i W_\mathrm{L}^2}{\sqrt{2}}, \nonumber \\&\!\!\!\left( \begin{array}{c} A_\mathrm{L} \\ Z_\mathrm{L} \end{array} \right) = \left( \begin{array}{r@{\quad }c} \cos \theta _\mathrm{W} &{} \sin \theta _\mathrm{W} \\ -\sin \theta _\mathrm{W} &{} \cos \theta _\mathrm{W} \end{array} \right) \left( \begin{array}{c} B_\mathrm{L} \\ W_\mathrm{L}^3 \end{array} \right) , \nonumber \\&\!\!\!W_\mathrm{H}^{\pm } = \frac{W_\mathrm{H}^1 \mp i W_\mathrm{H}^2}{\sqrt{2}}, \nonumber \\&\!\!\!\left( \begin{array}{c} A_\mathrm{H} \\ Z_\mathrm{H} \end{array} \right) = \left( \begin{array}{c@{\quad }r} \cos \theta _\mathrm{H} &{} -\sin \theta _\mathrm{H} \\ \sin \theta _\mathrm{H} &{} \cos \theta _\mathrm{H} \end{array} \right) \left( \begin{array}{c} B_\mathrm{H} \\ W_\mathrm{H}^3 \end{array} \right) , \end{aligned}$$
(7)

where \(\theta _{\mathrm{W}}\) is the usual Weinberg angle and \(\theta _\mathrm{H}\) is the mixing angle defined by

$$\begin{aligned} \sin {\theta _{\mathrm{H}}} \simeq \frac{5 g g^{\prime }}{4(5 g^2 - g^{\prime 2})} \frac{v_{\mathrm{SM}}^2}{f^2}. \end{aligned}$$
(8)

At \(\mathcal O(v^{2}/f^{2})\) in the expansion of the Lagrangian (3), the mass spectrum of the gauge bosons after EWSB is given by

$$\begin{aligned} M_{W_{\mathrm{L}}}&= \frac{gv}{2}\left( 1-\frac{v^{2}}{12f^{2}}\right) ,\nonumber \\ M_{Z_{\mathrm{L}}}&= \frac{gv}{2\cos \theta _{\mathrm{W}}}\left( 1-\frac{v^{2}}{12f^{2}}\right) ,~~M_{A_{\mathrm{L}}}=0, \end{aligned}$$
(9)
$$\begin{aligned} M_{W_{\mathrm{H}}}&= M_{Z_{\mathrm{H}}}=gf\left( 1-\frac{v^{2}}{8f^{2}}\right) ,\nonumber \\ M_{A_{\mathrm{H}}}&= \frac{g'f}{\sqrt{5}} \left( 1-\frac{5v^{2}}{8f^{2}}\right) , \end{aligned}$$
(10)

where \(v=v_{\mathrm{SM}}\left( 1+\frac{1}{12}\frac{v_{\mathrm{SM}}^2}{f^2}\right) \) and \(v_{\mathrm{SM}}= 246\) 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 [29], \(V_{\mathrm{CW}}\), whose explicit form can be obtained after expanding the \(\Sigma \) field,

$$\begin{aligned} V_{\mathrm{CW}}&= \lambda _{\phi ^2} f^2 \mathrm{Tr}|\phi |^2 + i \lambda _{H \phi H} f ( H \phi ^\dagger H^T - H^* \phi H^\dagger )\nonumber \\&-\, \mu ^2 |H|^2 + \lambda _{H^4} |H|^4, \end{aligned}$$
(11)

where \(\lambda _{\phi ^2}\), \(\lambda _{H \phi H}\) and \(\lambda _{H^4}\) depend on the fundamental parameters of the model, whereas \(\mu ^2\), 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 \(HZZ\), \(HWW\) and \(HHZZ\), \(HHWW\) couplings involved in our calculations are modified at \(\mathcal O(v^{2}/f^{2})\), which are given by

$$\begin{aligned}&V_{HZ_{\mu }Z_{\nu }}=\frac{e^2v}{6\cos ^2\theta _{W}\sin ^2\theta _{W}}\left( 3-\frac{v^2}{f^2}\right) g_{\mu \nu },\end{aligned}$$
(12)
$$\begin{aligned}&V_{HW_{\mu }W_{\nu }}=\frac{e^2v}{6\sin ^2\theta _{W}}\left( 3-\frac{v^2}{f^2}\right) g_{\mu \nu },\end{aligned}$$
(13)
$$\begin{aligned}&V_{HHZ_{\mu }Z_{\nu }}=\frac{e^2}{2\cos ^2\theta _{W}\sin ^2\theta _{W}}\left( 1-\frac{v^2}{f^2}\right) g_{\mu \nu },\end{aligned}$$
(14)
$$\begin{aligned}&V_{HHW_{\mu }W_{\nu }}=\frac{e^2}{2\sin ^2\theta _{W}}\left( 1-\frac{v^2}{f^2}\right) g_{\mu \nu }. \end{aligned}$$
(15)

3 Calculation and numerical results

In our numerical calculations, the SM parameters are taken as follows [30]:

$$\begin{aligned} \nonumber&G_{F}=1.16637\times 10^{-5}\,\mathrm{GeV}^{-2}, \quad \sin ^{2}\theta _{\mathrm{W}}=0.231,\\&\alpha _{e}=1/128,\quad m_{{H}}=125~\mathrm{GeV},\\ \nonumber&M_{Z}=91.1876~\mathrm{GeV},\quad m_{b}=4.65\,\mathrm{GeV},\quad m_{e}=0.51\,\mathrm{MeV},\\&m_{\mu }=105.66\,\mathrm{MeV}. \end{aligned}$$

According to the constraints in Refs. [31–34], we require the scale to vary in the range \(500 \mathrm{GeV} \le f\le 1,500\) GeV.

At the tree level, the Feynman diagrams relevant to the process \(e^{+}e^{-}\rightarrow ZHH\) and the process \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH(\nu =\nu _{e},\nu _{\mu },\nu _{\tau })\) are showed in Figs. 1 and 2, respectively. In both processes, we can see that only the first column of the diagrams, i.e. Figs. 1a, 2a and d, are the signal diagrams which involve the Higgs trilinear self-coupling vertex \(HHH\); the other ones are irreducible background diagrams. For the process \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\), the \(ZZ\)-fusion process is equally or even more important compared with the \(WW\)-fusion process at the lower center-of-mass energy.

Fig. 1
figure 1

Feynman diagrams for \(e^{+}e^{-}\rightarrow ZHH\) at the tree level

Full size image
Fig. 2
figure 2

Feynman diagrams for \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) at the tree level

Full size image

On the left panel of Fig. 3, we show the dependence of the production cross sections \(\sigma \) of the processes \(e^{+}e^{-}\rightarrow ZHH\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) on the center-of-mass energy \(\sqrt{s}\) for the scale \(f=700\) GeV in the LHT model and the SM, respectively. We can see that the \(e^{+}e^{-}\rightarrow ZHH\) cross section decreases with increasing center-of-mass energy \(\sqrt{s}\) while \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) cross section increases. The \(e^{+}e^{-}\rightarrow ZHH\) cross section has the peak value around \(\sqrt{s}\sim 500\) GeV. For \(\sqrt{s}\sim 1\) TeV, the two cross sections are of the same order of magnitude, with \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) being the larger source of Higgs boson pairs for \(\sqrt{s}\ge 1\) TeV. Since the \(\nu \bar{\nu }HH\) production is peaked in the forward region, it is important to ensure that an efficient tagging of the \(HH \rightarrow b\bar{b}b\bar{b}, W^{+}W^{-}W^{+}W^{-}\) decay can be achieved.

Fig. 3
figure 3

The production cross sections \(\sigma \) versus the center-of-mass energy \(\sqrt{s}\) for \(f=700\) GeV (left) and the relative corrections \(\delta \sigma /\sigma \) versus the scale \(f\) for \(\sqrt{s}=500\) GeV (right)

Full size image

On the right panel of Fig. 3, we show the dependence of the relative corrections \(\delta \sigma /\sigma \) of the processes \(e^{+}e^{-}\rightarrow ZHH\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) on the scale \(f\) for the center-of-mass energy \(\sqrt{s}=500\) GeV. We can see that the relative correction \(\delta \sigma /\sigma \) 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 [35], the relative correction \(\delta \sigma /\sigma \) of the process \(e^{+}e^{-}\rightarrow ZHH\) can reach \(-30\) to \(-25\,\%\) and the relative correction \(\delta \sigma /\sigma \) of the process \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) can reach \(-16\) to \(-12\,\%\) for the scale \(f\) in the range \(600\)–\(700\,\mathrm{GeV}\). 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.

In the following calculations, we will study the process \(e^{+}e^{-}\rightarrow ZHH\) through the \((l\bar{l})(b\bar{b})(b\bar{b})\) mode and process \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) through \(\nu \bar{\nu }(b\bar{b})(b\bar{b})\) mode. We generate the parton-level signal and background events with MadGraph5 [36].

3.1 \(e^{+}e^{-}\rightarrow ZHH\rightarrow (l\bar{l})(b\bar{b})(b\bar{b})\)

For light Higgs boson masses, the Higgs boson decays predominantly in a \(b\bar{b}\) pair. The \(ZHH\rightarrow q\bar{q}b\bar{b}b\bar{b}\) final state benefits from a high statistics with \(\sim 35\,\%\) of the final states but requires a more complicated analysis. By contrast, though \(ZHH\rightarrow l\bar{l}b\bar{b}b\bar{b}(l=e,\mu )\) represents only \(\sim 5\,\%\) of the total final state, this topology produces an easy signature. Therefore, we choose the \(l\bar{l}b\bar{b}b\bar{b}\) 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 \(l\bar{l}\) with invariant \(Z\) mass.

In Fig. 4 we display the invariant mass of four \(b\)-jets \(M_{HH}\) in the SM and LHT model. The \(M_{HH}\) distribution is well 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 which \(\bar{b}\)-jet when reconstructing the Higgs bosons in the event, here we give the four \(b\)-jets invariant mass distribution \(M_{HH}\).

Fig. 4
figure 4

Normalized \(M_{HH}\) distributions in the SM and the LHT through the production of \(e^{+}e^{-}\rightarrow ZHH\rightarrow l\bar{l}b\bar{b}b\bar{b}\) for \(\sqrt{s}=500\) GeV, \(f=700\) GeV

Full size image

The background events mainly come from \(e^{+}e^{-}\rightarrow ZZZ \rightarrow (l\bar{l})(b\bar{b})(b\bar{b})\) and \(e^{+}e^{-}\rightarrow ZZH \rightarrow (l\bar{l})(b\bar{b})(b\bar{b})\).

In Fig. 5, we display the total transverse energy \(H_{T}\) and the transverse momentum \(p_{T}^{l}\) distributions of \((l\bar{l})(b\bar{b})(b\bar{b})\) in the signal with \(f=700\) GeV and backgrounds for \(\sqrt{s}=500\) GeV. According to Fig. 5, we can impose the cut \(H_{T} < 450\) GeV to suppress the backgrounds. However, due to such the same parton-level final states as the signal, we need more complicated technique and more careful analysis to distinguish the signal and the backgrounds.

Fig. 5
figure 5

Normalized distributions of \(H_{T}\) and \(p_{T}^{l}\) in the signal with \(f=700\) GeV and backgrounds for \(\sqrt{s}=500\) GeV

Full size image

3.2 \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\rightarrow \nu \bar{\nu }(b\bar{b})(b\bar{b})\)

Due to the dominant decay mode of Higgs is \(H\rightarrow b\bar{b}\), the experimental signature for \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) is then four \(b\)-jets (two pairs with invariant Higgs mass) plus missing energy and momentum. The dominant background \(\nu \nu bbbb\) mainly comes from \(e^{+}e^{-}\,{\rightarrow }\, ZZZ\) and \(ZZH\). Likewise, we display the invariant mass distribution \(M_{HH}\) of the four \(b\)-jets in Fig. 6.

Fig. 6
figure 6

Normalized \(M_{HH}\) distributions in the SM and the LHT through the production of \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\rightarrow \nu \bar{\nu }b\bar{b}b\bar{b}\) for \(\sqrt{s}=500\) GeV, \(f=700\) GeV

Full size image

In Fig. 7, we display the total transverse energy \(H_{T}\) and the missing energy \(\not \!\!\!E_{T}\) distributions of \(\nu \bar{\nu }b\bar{b}b\bar{b}\) in the signal with \(f=700\) GeV and backgrounds for \(\sqrt{s}=500\) GeV. According to Fig. 7, we can impose the cut \(H_{T} > 300\) GeV to suppress the backgrounds.

Fig. 7
figure 7

Normalized distributions of \(H_{T}\) and \(\not \!\!\!E_{T}\) in the signal with \(f=700\) GeV and backgrounds for \(\sqrt{s}=500\) GeV

Full size image

4 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 \(e^{+}e^{-}\rightarrow ZHH\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) have been investigated. For \(\sqrt{s}=500\) GeV, we calculated the production cross section and found that the relative correction of the process \(e^{+}e^{-}\rightarrow ZHH\) can reach \(-30\,\%\) and the relative correction \(\delta \sigma /\sigma \) of the process \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\) 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 \(e^{+}e^{-}\rightarrow ZHH \rightarrow l\bar{l} b\bar{b}b\bar{b}\) and \(e^{+}e^{-}\rightarrow \nu \bar{\nu }HH\rightarrow \nu \bar{\nu }b\bar{b}b\bar{b}\) were studied and some distributions of the signal and background were presented. Due to there is only slight difference between the signals and backgrounds, more complicated technique and more careful analysis are needed to distinguish them.