A fixed smooth solid sphere has centre \(O\) and radius \(a\). A particle of mass \(m\) is projected downwards with speed \(\sqrt{\frac{1}{6} a g}\) from the point \(A\) on the surface of the sphere, where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\) (see diagram). The particle moves in part of a vertical circle on the surface of the sphere. It loses contact with the sphere at the point \(B\), where \(O B\) makes an angle \(\beta\) with the upward vertical through \(O\).

Given that \(\cos \alpha=\frac{2}{3}\), find the value of \(\cos \beta\).