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 O-6) 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 (Lambda) 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 Lambda, the lower the phase transition strength and the larger the Wilson parameter, the wider the domain of Lambda. 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 O-6 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.