Let C be the circle formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the plane $z = - 1$. When a plane $\alpha$ containing the $x$-axis intersects the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ to form a circle that meets C at exactly one point, and a normal vector to plane $\alpha$ is $\vec { n } = ( a , 3 , b )$, find the value of $a ^ { 2 } + b ^ { 2 }$. [4 points]