The Renormalized Thermal Mass with Non-Zero Charge Density

The linear $\delta$ expansion is used to obtain corrections up to O$({\delta }^2)$ to the self-energy for a complex scalar field theory with a $\lambda ({\varphi }^{\star }\varphi {)}^2$ interaction at high temperature and non-zero charge density. The calculation is done in the imaginary-time formalism via the Hamiltonian form of the path integral. Nonperturbative results are generated by a systematic order by order variational procedure and the dependence of the critical temperature on the chemical potential $\mu$ is obtained.