Hugenholtz -- Pines relations and the critical temperature of a Rabi coupled binary Bose system

Using a theoretical field Gaussian approximation we have studied Rabi coupled binary Bose system at low temperatures. We have derived extended Hugenholtz - Pines relations taking into account one body interaction (e.g. Rabi coupling) and studied the critical temperature $T_c$ of Bose-Einstein condensate transition. We have shown that, the shift of $T_c$ due to this interaction can not exceed $\sim 60\mathrm{\%}$ and goes to a plateau with increasing the parameter ${\mathrm{\Omega }}_R\mathrm{/}{T}_c^0$, where ${\mathrm{\Omega }}_R$ is the intensity of the coupling and $T_c^0$ is the critical temperature of the system with ${\mathrm{\Omega }}_R=0$. Moreover, the shift is always positive and does not depend on the sign of the one body interaction.