Functional integrals and phase stability properties in the $O(N)$ vector field condensation model

Using condensation of auxiliary Bose fields and the functional integral method, we derive an effective action of the binary $O(N)$ vector field model on a sphere. We analyze two models with different forms of the coupling constants: the binary field model on $S^3$ and the two-component vector field model on $S^d$. In both models, we obtain the convergence conditions for the partition function from the traces of a free propagator. From analytic solutions of the saddle-point equations, we derive phase stability conditions, which imply that the system allows the formation of coexisting condensates when the condensate densities of the complex Bose fields and the unit vector field satisfy a certain constraint. In addition, within the $1/N$ expansion of the free energy on $S^d$, we also find that the absolute value of free energy decreases as the dimension $d$ increases.