@David Roberts I posted the solution BEFORE Iosif posted his

@TanyaVladi I see it now, A function $\varphi$ is completely monotone on $[0, \infty)$ if and only if $\Phi=$ $\varphi\left(\|\cdot\|^2\right)$ is positive definite and radial on $\mathbb{R}^s$ for all $s$, since the composition of completely monotone and Bernstein is completely monotone we are back to positive definite.

math.iit.edu/~fass/603_ch2.pdf

Since $ e^{-r^{2\gamma} t^2}$ can be represented as a Gaussian mixture, $\phi(r^{\gamma})$ is radial