Sphaleron in the first-order electroweak phase transition with the dimension-six Higgs field operator

By adding the dimension-six operator for the Higgs potential (denoted O₆) in Standard Model, we have a first-order electroweak phase transition (EWPT) whose strength is larger than unity. The cutoff parameter of the dimension-six Higgs operator (Λ ) is found to be in the range 593-860 GeV with the Wilson parameter equal to unity; it is also shown that the greater the Λ , the lower the phase transition strength and the larger the Wilson parameter, the wider the domain of Λ . At zero temperature, the sphaleron energy is calculated with a smooth Ansatz and an Ansatz with scale-free parameters, thereby we find that smooth profiles are not more accurate than profiles with scale-free parameters. Then, using the one-loop effective Higgs potential with the inclusion of O₆ instead of all possible dimension-six operators, we directly calculate the electroweak sphaleron energy at finite temperature with the scale-free parameters Ansatz and show that the decoupling condition is satisfied during the phase transition. Moreover, we can reevaluate the upper bound of the cutoff scale inferred from the first-order phase transition. In addition, with the upper bound of the cutoff parameter (about 800-860 GeV), EWPT is a solution to the energy scale of the dimension-six operators. There is an extended conclusion that EWPT can only be solved at a large energy scale than that of SM.