We study the 8D $U(1)$ case in the presence of an external scalar field that is a massless scalar field with the mass of the scalar field and is coupled to a $\mathbb{Z}_2$-vector. In this case, we compute the resting state curvature of the state space, in the presence of an external scalar field, and we determine that the resting state curvature is given by the rate of the resting state decay.