Long time behavior for derivative nonlinear wave equation on the high-dimensional torus

In this paper, we study the derivative nonlinear wave equation$ u_{tt}-\Delta u+mu+\Lambda^{\alpha}f(\Lambda^{\alpha}u) = 0,\quad x\in\mathbb{T}^{d}: = (\mathbb{R}/2\pi\mathbb{Z})^{d} $with small-amplitude initial data of size $ \varepsilon $ in Sobolev spaces, where $ \frac{1}{2} < \alpha\leq\frac{3}{4} $, the operator $ \Lambda = (-\Delta+m)^{\frac{1}{2}} $ with $ m > 0 $, and $ f $ is an analytic function of order at least two at the origin. After performing any finite steps of Birkhoff normal form process and one step of vector field transformation, we establish the long time behavior of solutions. Due to the unboundedness of the nonlinear vector field, we provide different coefficient estimates and energy estimates tailored to the low and high frequencies. We emphasize that after the Birkhoff normal form process, certain terms are still not controlled via the standard energy estimates. We therefore construct a vector field transformation to eliminate these terms, while ensuring that all newly generated terms are bounded.