1. Introduction

In theories with a simple scalar field and a Z ₂ invariant selfinteraction potential, it is possible to find stable topological kinks interpolating asymptotically between the minima of the system. These solutions are interesting in the framework of the gravitational theories with extra dimensions because the kink or domain wall interpolates between two Anti de Sitter spacetimes and induces the standard gravitational interaction on the four-dimensional sector of the warped structure [¹-⁴]. Such scenarios are referred to as brane-worlds, and it is surmised that the standard model fields should be localized on the topological defect [⁵-⁷].

As a first approximation to a scenario where the symmetry group of the standard model can be recovered inside the domain wall, in absence of gravity but in presence of a nonabelian symmetry, kink solutions have been obtained in several opportunities [⁸-¹⁰]. In Ref. [8] three kink solutions for a SO(10) theory inducing asymptotically the breaking symmetry SO(10) → SU(5) were determined; subsequently, in Ref. [¹⁰] the local stability of these scenarios was evaluated finding in two of them tachyonic Poschl-Teller modes in the¨ spectrum of scalar perturbations. Other models in terms of E ₆ group were discussed in Ref. [⁹].

Among the non-abelian solutions we highlights the one where the kink interpolates between SU(3)×SU(2)×U(1) vacuum expectation values (vev) of a SU(5) × Z ₂ invariant potential [¹¹-¹⁴]. Since this issue is the focus of this paper, let us review in detail this scenario.

Consider the bosonic sector of the SU(5) model in (1+1) dimensions

with Φ a scalar field transforming in the adjoint representation of the symmetry group

where T _j , j = 1,...,24, are traceless hermitian generators of SU(5). In the potential (2), µ, h and λ are the parameters of the theory and V ₀ is a constant to adjust conveniently the minimum to zero.

It is well know that there are two possible non-trivial form for the minimum of (2) [¹⁵]:

and

which lead to the breaking patterns

and

respectively.

Due to the absence of cubic terms in (2) the Z ₂ symmetry is included in the model and kink solutions are expected, such that

where U is an element of SU(5) and Φ depends only on the coordinate z. In fact, for h = −3λ/20 and λ > 0, the symmetry breaking pattern

can be induced asymptotically by the following non-abelian kink

which, at z = ±∞ goes to

Notice that (12) is compatible with the constraint (8) and that, from trace properties, the vacuums (4) and (11) are equivalent for the scalar potential. On the other hand, in accordance with (11) and (12), SU(3)×SU(2)×U(1) is embedded asymptotically in SU(5) in different ways. Moreover, inside the kink, Φ _A(z = 0) ∼ diag(1,−1,0,1,−1), the unbroken group is

This solution, as well as its perturbative stability, were determined in [¹¹]; subsequently, (10) was recovered as a particular case of a kink in SU(N) × Z ₂ [¹²], the extension to curved spacetime in five dimensions was found in [13,¹⁴] where (10) induces the symmetry group of the standard model at the boundary of an AdS5 warped spacetime.

With respect to the vev (5), curiously, up to now, a kink solution for the model (1) has not been reported; however, a self-gravitating kink inducing unbroken group (7) was obtained in [¹³]. Considering the absence of a flat kink for (5), our proposal for this paper is to find this solution and evaluate its stability under small perturbations.

2. Kink solution

Let us consider a kink solution, Φ _B, for (1) in correspondence, at infinity, with the symmetry breaking pattern

For this case, it is convenient to write the non-abelian field in terms of diagonal generators of SU(5)

where

The unbroken group (14) is induced by a kink when (15) satisfy the boundary conditions

The equations of motion for the coefficients of (15) are given by

where prime means derivative with respect to z. Suggested by (20, 21) we choose conveniently φ ₁ = φ ₂ = 0; thus, we obtain a pair of coupled equations for φ ₃ and φ ₄, namely

which can be decouple for λ = −10h/9, h > 0. Now, for the remaining equations we require

thus, we find the solution

such that at z = ±∞, the unbroken group (14) is embedded in SU(5) in different ways. On the other hand, in the core of the kink, z = 0, the remaining symmetry group is

3. Perturbative stability

Now, in order to study the perturbative stability of (29) let us consider, in the energy of the system, small deviations from kink solution Z

where Ψ = ψ _j T _j . Thus, to second order in ϵ (the term proportional to ϵ is zero via the equation of motion of Φ_B), we find that

where

which is diagonal and, hence, the double sum in (32) can be written as follows

and the stability problem consists in determining the eigenvalue associated with each generator of SU(5). Fortunately, for several generators we obtain the same eigenvalues equation and, after group them, only five non trivial cases need to be checked. For the trivial cases, j = 19,20, whose generators are broken everywhere, a vanishing potential is obtained.

For the remaining five cases it is convenient to consider ξ=μz∕2, to make dimensionless the equations of motions. For the set of generators labelled by j = 1,...,6, 21,22, basis for SU(3) in (30), we get

which can be rewritten in terms of a Poschl-Teller potential,¨ whose bound states are well known [¹⁶]. Thus, the spectrum of scalar states is determined for all j by a pair of normalizable eigenfunctions

and a set of free states from m2>37μ2∕8.

In the cases identified with j ₊ = 7,...,12 and j ₋ = 13,...,18, broken generators with respect to (30) but unbroken with respect to (14) at ξ →±∞, we have

In this case the Schrödinger-like equation is associated to a non-conventional potential (see Fig. 1) which exhibits a negative well in −∞ < ξ ≤ 0 for ψ ₊. For ψ ₋ the potential has an equivalent profile, but in the region ∞ > ξ ≥ 0. However, the eigenvalues of the equation are positive since (38) can be factorized as follows [¹⁷]

where

In addition, the zero mode can be obtained, ψ ₀ ∼ 1∓tanhξ, which is out of the spectrum of eigenfunctions because it is not normalizable, as expected in concordance with Fig. 1.

FIGURE 1 Plot of the potential for the scalar perturbations ψ _j+ along the broken generators j ₊ = 7,...,12. The potential profile for ψ _j − with j ₋ = 13,...,18 is a mirror image of the shown one. 

Finally, in the scalar field directions T ₂₃ and T ₂₄, we find

with eigenvalues bounded as m232>μ2 and m242≥0. In the last case, two localized states are found

being the first one, (43), the translational mode of the flat space SU(5) × Z₂ kink [¹¹].

Since in all cases the eigenvalues are positive, the perturbations do not induce instability on (29) and hence the non-abelian flat domain wall Φ_B is a locally stable solution of (1, 2).

4. Ending comments

We have derived a flat SU(5)×Z₂ kink interpolating asymptotically between Minkowskian vacuums with the symmetry breaking pattern SU(5)×Z₂ → SU(4)×U(1)/Z₄ and with unbroken group SU(3)×U(1)×U(1)/Z₃ in the core of the wall.

With regard to the spectrum of scalar fluctuations, we do not find tachyonic modes compromising the local stability of the non-abelian Wall. In particular we find the translational zero mode and several Pöschl-Teller confined scalar states along SU(3) basis.

Non-abelian kinks as brane worlds is the next issue that we would like to study. The main problem is that we need to find a scenario where the symmetry group of the standard model corresponds to the unbroken symmetry inside the kink. In our opinion, solutions similar to (10) and (29) are a first approximation to this open problem.

As commented in the introduction, another option has been already explored in [8, 9] where the symmetry of the theory is determined by the SO(10) group. In this case, the unbroken symmetry for finite z is achieved used the clash-ofsymmetries mechanism. Thus, SU(3)× SU(2)×U(1)× U(1) may be obtained in the core of a SO(10) wall, which is almost the symmetry expected for a more realistic model.