Failure of the naïve loop expansion for the effective potential in φ4 field theory when there is “broken symmetry”

We determine the effective potential Vφ] in zero dimensional gφ⁴ field theory when there is broken symmetry classically $\text{V}{}_{\mathrm{<mtext>cl</mtext>}}\text{=}\text{1}\text{4}\text{g(φ}{}^2\text{? a}{}^2\text{)}{}^2\text{]}$ as a power series in 1/g. We find that the usual loop expansion ignores a secondary minimum of the path integral and is only valid for |φ| ? a.When |ja| ? 1, where j is the external source, there is a secondary minimum of equal importance to the absolute minimum. In that region a kink solution to the differential equation for φj] exists, φj] = a tanh(ja) + O(ja²/ga⁴), which interpolates between the minima at φ = ?a and φ = +a. We obtain the 1/g expansion for Vφ] for φ < a and φ ? a. For all g, a mean-field approximation gives an expression for Vφ] which interpolates between these two regimes. In higher dimensions the kink solution becomes $\text{φj] = a}\text{tanh}\text{(jaΩ)}$, where Ω is the volume of space-time, suggesting a discontinuity in φj] in the infinite-volume limit.