Portal de Periódicos da CAPES

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

1983; Elsevier BV; Volume: 224; Issue: 3 Linguagem: Inglês

10.1016/0550-3213(83)90383-8

1873-1562

Carl M. Bender, Fred Cooper,

Magnetic confinement fusion research

We determine the effective potential V[φ] in zero dimensional gφ4 field theory when there is broken symmetry classically [Vcl = 14g(φ2 − a2)2] 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 φ[j] exists, φ[j] = a tanh(ja) + O(ja2/ga4), which interpolates between the minima at φ = −a and φ = +a. We obtain the 1/g expansion for V[φ] for φ < a and φ ⩾ a. For all g, a mean-field approximation gives an expression for V[φ] which interpolates between these two regimes. In higher dimensions the kink solution becomes φ[j] = a tanh(jaΩ), where Ω is the volume of space-time, suggesting a discontinuity in φ[j] in the infinite-volume limit.