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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.65 n.1 México Jan./Feb. 2019  Epub Nov 09, 2019

 

Investigación

A SU(5)×Z2 kink solution and its local stability

R. Guerreroa 

R. O. Rodrigueza 

R. Chavezb 

a Facultad de Ciencias, Escuela Superior Politécnica de Chimborazo, EC060155-Riobamba, Ecuador.

bDepartamento de Ciencias Básicas, Universidad Politécnica Salesiana, 170105-Quito, Ecuador.

Abstract

A non-abelian kink inducing asymptotically the breaking pattern SU(5)×Z 2SU(4)×U(1)/Z 4 is obtained. We consider a fourth order Higgs potential in a 1+1 theory where the scalar field is in the adjoint representation of SU(5). The perturbative stability of the kink is also evaluated. A Schrodinger-like equation for the excitations along each¨ SU(5) generator is determined, and in none of the cases negative eigenvalues compromising the stability of solution are found. In particular, several bounded scalar states are determined, being one of them the translational zero mode of the flat space SU(5)× Z 2 kink.

Keywords: SU(5) kink; local stability

PACS: 11.27.+d; 04.50.-h

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

L=-Tr(mΦmΦ)-V(Φ), (1)

V(Φ)=-μ2Tr(Φ2)+h(Tr(Φ2))2+λTr(Φ4)+V0 (2)

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

ΦUΦU,    U=exp(iαjTj),Tr(Tj1Tj2)=12δj1j2 (3)

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]:

<ΦA>diag(2,2,2,-3,-3),          λ>0, (4)

and

<ΦB>diag(1,1,1,1,-4),          λ<0; (5)

which lead to the breaking patterns

SU(5)SU(3)×SU(2)×U(1) (6)

and

SU(5)SU(4)×U(1), (7)

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

Φ(z=-)=-UΦ(z=+)U, (8)

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

SU(5)×Z2SU(3)×SU(2)×U(1)Z3×Z2 (9)

can be induced asymptotically by the following non-abelian kink

ΦA=52μλ[diag(1,-1,0,1,-1)+15tanh(μz2)diag(-1,-1,4,-1,-1)]. (10)

which, at z = ±∞ goes to

ΦA(z=+)=15μλdiag(2,-3,2,2,-3) (11)

ΦA(z=-)=15μλdiag(3,-2,-2,3,-2). (12)

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

SU(2)×SU(2)×U(1)×U(1)Z2×Z2. (13)

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

SU(5)×Z2SU(4)×U(1)Z4. (14)

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

Φ=ϕ1T21+ϕ2T22+ϕ3T23+ϕ4T24 (15)

where

T21=12diag(1,-1,0,0,0), (16)

T22=123diag(1,1,-2,0,0), (17)

T23=12diag(0,0,0,1,-1), (18)

T24=1215diag(2,2,2,-3,-3). (19)

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

Φ(z=+)=v(T23+35T24)=<ΦB>, (20)

Φ(z=-)=v(T23-35T24)=-U<ΦB>U. (21)

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

ϕ1''=-[μ2-(h+2λ5)ϕ42-(h+λ2)(ϕ12+ϕ22)-hϕ32]ϕ1+2λ5ϕ1ϕ2ϕ4, (22)

ϕ2''=-[μ2-(h+2λ5)ϕ42-(h+λ2)(ϕ12+ϕ22)-hϕ32]ϕ2+λ5ϕ4(ϕ12-ϕ22), (23)

ϕ3''=-[μ2-(h+9λ10)ϕ42-(h+λ2)ϕ32-h(ϕ12+ϕ22)]ϕ3, (24)

ϕ4''=-[μ2-(h+7λ30)ϕ42-(h+2λ5)(ϕ12+ϕ22)-(h+9λ10)ϕ32]ϕ4+λ5ϕ2(ϕ12-ϕ223). (25)

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

ϕ3''=[-μ2+(h+9λ10)ϕ42+(h+λ2)ϕ32]ϕ3, (26)

ϕ4''=[-μ2+(h+7λ30)ϕ42+(h+9λ10)ϕ32]ϕ4, (27)

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

ϕ3(z=±)=v,          ϕ4(z=±)=±35v; (28)

thus, we find the solution

ΦB=v[T23+35tanh(μz2)T24],v=3μ2h (29)

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

SU(3)×U(1)×U(1)Z3. (30)

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

E=dz[Tr(zΦB+ϵ zΨ)2+V(ΦB+ϵΨ)],ϵ1, (31)

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 ΦB), we find that

E=E[ΦB]+ϵ2dz ψj1(-δj1j2z2+Vj1j2)ψj2+O(ϵ3) (32)

where

Vj1j2(ΦB)=-12μ2δj1j2+hTr(ΦB2)δj1j2+4hTr(ΦBTj1)Tr(ΦBTj2)-409hTr(ΦB2Tj1Tj2)-209hTr(ΦBTj1ΦBTj2) (33)

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

(-12δj1j2z2+Vj1j2)ψj2=mj1j22ψj22, (34)

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=μz2, 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

[-12ξ2+14(5+3tanh2ξ)]ψj=2mj2μ2ψj, (35)

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

m02=4+78μ2,          ψ0cosh(1-7)/2(ξ) (36)

m12=378μ2,          ψ1cosh(1-7)/2(ξ)sinh(ξ) (37)

and a set of free states from m2>37μ28.

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

[-12 ξ2+tanhξ(tanhξ±1)]ψj±=2mj±2μ2ψj±. (38)

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]

(-ξ+β±)(ξ+β±)ψj±=4mj±2μ2ψj±, (39)

where

β±=21+3cosh2ξsinh2ξ+2(tanhξ1)ξ4+e±2ξ+e2ξ(3±4ξ). (40)

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

(-12ξ2+2)ψ23=2m232μ2ψ23 (41)

(-12ξ2+3tanh2ξ-1)ψ24=2m242μ2ψ24, (42)

with eigenvalues bounded as m232>μ2 and m2420. In the last case, two localized states are found

m02=0,          ψ0cosh-2(ξ) (43)

m12=34μ2,          ψ1cosh-2(ξ)sinh(ξ), (44)

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 ΦB 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.

Acknowledgements

We wish to thank Adriana Araujo for her collaboration to complete this paper. This work was partially financed by UPS project. R. Guerrero and R. O. Rodriguez wish to thank ESPOCH for hospitality during the completion of this work.

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Received: August 08, 2018; Accepted: September 26, 2018

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