Quadratic form of a difference of normals

Suppose that $X$ and $Y$ are independent and identically distributed $N_p(\mu, \Sigma)$ random vectors, where $\mu \in \mathbb{R}^p$ and $\Sigma$ is a positive definite matrix. Let $\chi_m^2$ denote chi-square distribution with $m$-degrees of freedom. Then which of the following statements is correct?