Winding Number Transition at Finite Temperature : Mottola-Wipf model with and without Skyrme term

The winding number transition in the Mottola-Wipf model with and without Skyrme term is examined. For the model with Skyrme term the number of discrete modes of the fluctuation operator around sphaleron is shown to be dependent on the value of $\lambda m^2$. Following Gorokhov and Blatter we derive a sufficient condition for the sharp first-order transition, which indicates that first-order transition occurs when $0< \lambda m^2 < 0.0399$ and $2.148 < \lambda m^2$. In the intermediate region of $\lambda m^2$ the winding number transition is conjectured to be smooth second order. For the model without Skyrme term the winding number transition is always first order regardless of the value of parameter.