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.