Advances in High Energy Physics

Volume 2016 (2016), Article ID 5353267, 7 pages

http://dx.doi.org/10.1155/2016/5353267

## Spontaneous Symmetry Breaking in 5D Conformally Invariant Gravity

^{1}Department of Physics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea^{2}Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Republic of Korea

Received 4 May 2016; Revised 27 June 2016; Accepted 28 June 2016

Academic Editor: Barun Majumder

Copyright © 2016 Taeyoon Moon and Phillial Oh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We explore the possibility of the spontaneous symmetry breaking in 5D conformally invariant gravity, whose action consists of a scalar field nonminimally coupled to the curvature with its potential. Performing dimensional reduction via ADM decomposition, we find that the model allows an exact solution giving rise to the 4D Minkowski vacuum. Exploiting the conformal invariance with Gaussian warp factor, we show that it also admits a solution which implements the spontaneous breaking of conformal symmetry. We investigate its stability by performing the tensor perturbation and find the resulting system is described by the conformal quantum mechanics. Possible applications to the spontaneous symmetry breaking of time-translational symmetry along the dynamical fifth direction and the brane-world scenario are discussed.

#### 1. Introduction

Conformal symmetry is an important idea which has appeared in diverse area of physics, and its application to gravity has started with the idea that conformally invariant gravity in four dimensions (4D) [1–4] might result in a unified description of gravity and electromagnetism. The Einstein-Hilbert action of general relativity is not conformally invariant. In realizing the conformal invariance of this, a conformal scalar field is necessary [5, 6] in order to compensate the conformal transformation of the metric, and a quartic potential for the scalar field can be allowed. Its higher dimensional extensions are straightforward. In five dimensions (5D), conformal symmetry can be preserved with a fractional power potential [7, 8] for the scalar field. So far, it seems that little attention has been paid to the 5D conformal gravity with its fractional power potential. Such a potential renders a perturbative approach inaccessible, but nonperturbative treatment may reveal novel aspects. One can also construct a conformally invariant gravity with the Weyl tensor via -squared gravity, but we focus on the Einstein gravity with a conformal scalar.

If the scalar field spontaneously breaks the conformal invariance with a Planckian VEV, the theory reduces to the 4D Einstein gravity with a cosmological constant [9, 10]. On the other hand, the spontaneous symmetry breaking of Lorentz symmetry [11] or gauge symmetry [12] in 5D brane-world scenario [13–16] was studied, but little is known in the context of 5D conformal gravity. In this paper, we explore 5D conformal gravity with a conformal scalar and investigate possible consequences in view of the spontaneous symmetry breaking.

Let us consider 5D conformal scalar action of the form where is the five-dimensional curvature scalar, is the action for some matter^{1}, and , run over 0, 1, 2, 3, 4. Here, is a dimensionless parameter describing the nonminimal coupling of the scalar field to the spacetime curvature. Also a parameter with corresponds to canonical (ghost) scalar. For , the conformal invariance of action (1) without matter term forces to bewhere is a constant and the corresponding conformal transformation is given byWhen , the conformal scalar has a negative kinetic energy term, but we regard it as a gauge artifact [17] which can be eliminated from the beginning through field redefinition. Even with no scalar field remaining after gauging away for both cases (), the physical mass scale can be set since the corresponding vacuum solution requires introduction of a scale which characterizes the conformal symmetry breaking. In 4D conformal gravity, it is known that the conformal symmetry can be spontaneously broken at electroweak [10, 18] or Planck [9, 10] scale. In all cases, the action (1) becomes 5D Einstein action with a cosmological constant by redefinition of the metric, , but we stick to the above conformal form (1) of the action to argue with the spontaneous breakdown of the conformal symmetry.

The paper is organized as follows. In Section 2, we perform the dimensional reduction from five to four dimensions by using the ADM decomposition. In Section 3, we present exact solutions with four-dimensional Minkowski vacuum () and check if they can give a spontaneous breaking of the conformal symmetry. In Section 4, the gravitational perturbation and their stability for the solutions are considered. In Section 5, we include the summary and discussions.

#### 2. Dimensional Reduction (5D to 4D)

In order to derive the 4D action from the 5D conformal gravity (1), we make use of the following ADM decomposition where the metric in 5D can be written as^{2}To describe the background solution, we go to the “comoving” gauge and choose . In this case, we can recover our 4D spacetime by going onto a hypersurface , which is orthogonal to the 5D unit vector:along the extra dimension, and can be interpreted as the metric of the 4D spacetime. Using the metric ansatz (4), one obtainswhere the asterisk denotes the differentiation with respect to and is the four-dimensional Laplacian. Using this, we find that action (1) becomesOne can check that the above action (7) is invariant with respect to four-dimensional diffeomorphism with . It is also invariant under and apart from the matter action.

Before going further, we would like to comment on the homogeneous solution to the equations of motion given in 5D conformal gravity (1) without matter term. To this end, we first consider the Einstein equation for action (1), whose form is given by and the scalar equation can be written as where is 5D covariant derivative, , and the prime denotes the differentiation with respect to . One can easily check that the homogeneous solution to (8) and (10) is given byWe note that this solution can be approached in diverse ways. Firstly, field redefinition necessitates introduction of scale, which leads to five-dimensional Planck mass . Secondly, solution (11) corresponds to a gauge fixed case () through the conformal transformation (3). Lastly, it can be interpreted as a vacuum solution obtained when considering an effective potential (we will study the effective potential for details at the end of the next section). In all cases, conformal symmetry is spontaneously broken with a symmetry breaking scale ~. In addition, they yield the physically equivalent results: de Sitter or anti-de Sitter . It is to be noticed that both cases are classically stable. Also there is a huge degeneracy of vacuum solutions due to conformal invariance such that if is a solution, then (, ) is also a solution for an arbitrary function . In the next section, we will investigate the explicit solution form, starting from the reduced action (7) without matter term.

#### 3. Exact Solutions

From action (7) without matter term, we find the equation of motion for asand the equation for four-dimensional metric is given byAlso the equation of motion for the scalar field can be written asNow one can equate (12) and trace of (13) and then obtainSubstituting the above equation back into (12), we arrive at

Let us focus on the conformally invariant case with and . In order to find solutions for this case, we consider the following ansatz:where and are constants. For this ansatz, (15)–(17) becomeThe above equation (21) determines as a function of ; namely, each hypersurface has different values of . But, we will restrict our attention to four-dimensional Minkowski space. Then, we notice that (19)–(21) always allow trivial vacuum solution , , independent of and , in general. The search for nontrivial vacuum with is facilitated by the fact that the coefficients in (19) and (20) come out right so that the two equations are identical. Finally, for the 4D Minkowski vacuum (), we can obtain two solutions: (i) the of (19)–(21) vanishes when and , which yields , and in this case, is arbitrary; (ii) for and , they allow the solution of . We summarize the 4D Minkowski solutions as follows:

Now we turn to an issue related to the spontaneous breaking of the conformal symmetry. We notice that the spontaneous symmetry breaking can be realized for a negative value of the curvature scalar with and . To see this, we consider an effective potential for the canonical scalar field () and in action (1) as As was mentioned in (2), here is fixed as a negative value of for the canonical scalar , which preserves the conformal symmetry of action (1). Since solution (i) () with a stable equilibrium does not provide a symmetry-broken phase, we focus on the case (ii) with giving (hereafter we fix ).

It turns out that, for (ii) with the positive 5D curvature scalar , we have only one vacuum solution of , while for , there exist two vacua of with nonzero value:where is given by . In this case, the conformal symmetry is spontaneously broken with the symmetry breaking scale given by (24). This result is summarized in Figure 1 which shows that the effective potential with has only one minimum (a), while for , it has two minima with (b) which corresponds to the case of spontaneous breaking of the conformal symmetry.