Generalized Ellis-Bronnikov Wormhole Solution in the scalar-Einstein-Gauss-Bonnet 4d Gravitational Model

We consider the sEGB 4d gravitational model with a scalar field \(\varphi \left( u\right)\), Einstein and Gauss-Bonnet terms. The model action contains a potential term \(U\left( \varphi \right)\), a Gauss-Bonnet coupling function \(f\left( \varphi \right)\) and a parameter \(\varepsilon = \pm 1\), where \(\varepsilon = 1\) corresponds to the usual scalar field, and \(\varepsilon = -1\) to the phantom field. In this paper, the sEGB reconstruction procedure considered in our previous paper is applied to the metric of the Ellis-Bronnikov solution, which describes a massive wormhole in the model with a phantom field (and zero potential). For this metric, written in the Buchdal parameterization with a radial variable u, we find a solution of the master equation for \(f\left( \varphi \left( u\right) \right)\) with the integration (reconstruction) parameter \(C_0\). We also find expressions for \(U\left( \varphi \left( u\right) \right)\) and \(\varepsilon \dot{\varphi }^2 = h\left( u\right)\) for \(\varepsilon = \pm 1\). We prove that for all non-trivial values of the parameter \(C_0 \ne 0\) the function \(h\left( u\right)\) is not of constant sign for all admissible \(u \in \left( -\infty , +\infty \right)\). This means that for a fixed value of the parameter \(\varepsilon = \pm 1\) there is no non-trivial sEGB reconstruction in which the scalar field is a purely ordinary field (\(\varepsilon = 1\)) or a purely phantom field (\(\varepsilon = - 1\)).