A General Thick Brane Supported by a Scalar Field

A thick Z2 -symmetric domain wall supported by a scalar field with an arbitrary potential V (φ) in 5D general relativity is considered as a candidate brane world. We show that, under the global regularity requirement, such a configuration (i) has always an AdS asymptotic far from the brane, (ii) is only possible if V (φ) has an alternating sign and (ii) V (φ) satisfies a certain fine-tuning type equality. The thin brane limit is well defined and conforms to the Randall-Sundrum (RS2) brane world model if the asymptotic value of V (φ) (related to Λ, the effective cosmological constant) is kept thickness-independent. Universality of such a transition is demonstrated using as an example exact solutions for stepwise potentials of different shapes. One more result is that, due to scale invariance of the Einstein-scalar equations, any given regular solution creates a one-parameter family of solutions with different potentials. In such families, a thin brane limit does not exist while the ratio Λ/σ (σ is the brane tension) is thickness-independent and is in general different from its value in the RS2 model.