Stable static structures in models with higher-order derivatives

Abstract

We investigate the presence of static solutions in generalized models described by a real scalar field in four-dimensional space–time. We study models in which the scalar field engenders higher-order derivatives and spontaneous symmetry breaking, inducing the presence of domain walls. Despite the presence of higher-order derivatives, the models keep to equations of motion second-order differential equations, so we focus on the presence of first-order equations that help us to obtain analytical solutions and investigate linear stability on general grounds. We then illustrate the general results with some specific examples, showing that the domain wall may become compact and that the zero mode may split. Moreover, if the model is further generalized to include k-field behavior, it may contribute to split the static structure itself.

Introduction

In this work we deal with relativistic models described by a single real scalar field with generalized dynamics in four-dimensional space–time. The study is inspired on the Galileon field, which is a real scalar field that engenders Galilean invariance, that is, if $\pi =\pi \left(x\right)$ is real, it is a Galileon field if its Lagrange density is symmetric under the Galilean and shift transformation $\pi \to \pi +a\cdot x+b$, with $a$ being a constant vector and $b$ a constant scalar.

The Galileon field was studied in  [1], [2] aimed to investigate self-accelerating solutions in the absence of ghosts, and has been further investigated in a diversity of contexts, with direct phenomenological applications, as one can see in the recent reviews  [3], [4], [5]. In particular, in  [6], [7], [8], [9], [10], [11] the authors deal with solitonic solutions and supersymmetrization. In  [6] it is shown that the Galileon field cannot give rise to static solitonic solutions; however, in  [7] one investigates the presence of soliton-like traveling waves for the Galileon field in two-dimensional space–time. Also, in Refs.  [8], [9] the authors offer other interesting results on solitons and Galileons. In  [10], supersymmetry is implemented starting from ghost condensate theories  [12]; see also Refs.  [13], [14], [15] for other studies on supersymmetry, generalized models and integrability.

One motivation to study the Galileon field is inspired by the fact that the Galilean invariance is capable of inducing an important feature to the Galileon field, which keeps its equation of motion a second-order differential equation. This and the presence of supersymmetry suggest that we search for a first-order framework, that is, for first-order differential equation that solves the equation of motion. We shall do this, extending the model, using the Galileon field to control the kinematics, but adding other terms, which break the Galilean symmetry and allow for the presence of spontaneous symmetry breaking, giving rise to localized static solutions. We call the scalar field, generalized Galileon field. We remark that the Galilean symmetry forbids the appearance of static solutions  [6], so we are forced to generalize the model, to break the Galilean symmetry to study the appearance of nontrivial static structures. Another motivation comes from gravity: we know that minimal coupling of Galileons to gravity leads to equations of motion which have higher-order derivatives of the metric; however, this can be remedied with non-minimal couplings, at the expense of breaking the Galilean symmetry  [16].

Here we focus attention on the model

$$\mathcal{L}=K(\pi ,X)+F(\pi ,X)\square \pi \text{,}$$

in four-dimensional space–time. We consider that $K(\pi ,X)$ and $F(\pi ,X)$ are in principle arbitrary functions of $\pi$ and $X$, with $X$ being defined as

$$X=\frac{1}{2}{\partial }_{\mu }\pi {\partial }^{\mu }\pi \text{.}$$

We are using $\square \equiv g^{\alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }$, the metric is diagonal $(+,-,-,-)$ and the scalar field, space and time coordinates, and coupling constants are all dimensionless. Like in  [17], [18], we change the term ${\partial }_{\mu }\pi {\partial }^{\mu }\pi \square \pi$ to the more general form $F(\pi ,X)\square \pi$.

The generalization that we consider may break the Galilean symmetry, but the equation of motion preserves the second-order structure. We are interested in solutions of these theories in the presence of spontaneous symmetry breaking, and we shall search in particular for planar domain walls and for its classical stability. As one knows, domain walls are non-perturbative classical solutions which find applications in many areas in physics, describing transitions between disconnected states of minimum energy  [19], [20]. The main issue here is to study domain walls in models of scalar fields with generalized dynamics of the Galileon type. We may also include k-field dynamics  [21], as we have done before in Refs.  [22], [23], [24], [25], [26]. Here we focus on similar issues, with the scalar field now having generalized dynamics. The results show that the Galileon-like field may make the static solution compact, and may split the zero mode. Moreover, if we add generalized kinematics to the dynamical field, making the scalar field a generalized k-Galileon, the two contributions may contribute to split the static structure itself.

The investigation is organized as follows. In the next two sections we introduce the model and study linear stability on general grounds. We focus in particular on the first-order framework, where we search for first-order ordinary differential equations whose solutions also solve the equation of motion, which is second-order ordinary differential equation. In Section  4 we employ the method in order to investigate some distinct models explicitly, searching for static solutions and showing that they may engender interesting features. We end the work in Section  5, where we include our comments and conclusions.