Broken electroweak phase at high temperature in the littlest Higgs model with T parity

We have examined in detail the nonrestoration of symmetry at high temperature in a finite-temperature littlest Higgs model, without and with T parity, by evaluating the one-loop-order finite-temperature integrals of the effective potential numerically, without the high-temperature approximation T > m(i). We observe that in the model without T parity it is not possible to find a transition temperature within the allowed temperature range of the model (0 < T < 4f) if the UV completion factors are those which give the standard model electroweak minimum, as the effective potential always increases with temperature in the positive direction. However, in the case of the model with T parity, it is possible to find a transition temperature with the same set of UV completion factors, as, with the increase of temperature, the effective potential decreases in magnitude in the positive side, becomes negative, and increases in magnitude in the negative side, indicating symmetry breaking at high temperature. This type of symmetry nonrestoration at high temperature has been observed earlier in some models involving pseudo-Nambu-Goldstone bosons. The behavior of the global structure of the effective potential with T parity in the theory indicates a strong first-order electroweak phase transition, conducive to baryogenesis in the early Universe.