On the stability of the classical vacua in a minimal SU(5) five-dimensional supergravity model

We consider a five-dimensional supergravity model with SU(5) gauge symmetry and the minimal field content. Studying the arising scalar potential we find that the gauging of the U(1)R symmetry of the five-dimensional supergravity causes instabilities. Lifting the instabilities, the vacua are of anti-de-Sitter type and SU(5) is broken along with supersymmetry. Keeping the U(1)R ungauged the potential has flat directions along which supersymmetry is unbroken.


Introduction
It is well established that the standard model (SM) successfully describes all particle interactions at low energies. However, it is understood that the SM is an effective theory. At higher energies, the description of the elementary particle interactions demands a generalization of the SM.
The assumption of a unified description in terms of a renormalizable field theory, up to very high energies, leads to favourable generalizations, namely GUT theories [1], among which supersymmetric GUTs [2] play a central role. The consistent inclusion of gravity dictates that these generalizations should be effective descriptions of a more fundamental underlying theory, and string theory [3,4] is the most prominent candidate for this aim. Indeed from the ten-dimensional field theory, which is the effective point limit of the string theory, we can obtain, by suitable compactifications of the extra dimensions, consistent four-dimensional models compatible with the SM [5]. Along these lines it has been conjectured that one or two dimensions may be compactified at different scales, lower than the remaining ones [6]. Also, after developments concerning the duality symmetries of string theory and in the framework of 1.2 M-theory [7,8], the idea that our world may be a brane embedded in a higher-dimensional space has recently attracted much interest and has been studied intensively [9,10,11,12,13,14,15]. Besides the original compactifications, new possibilities have been proposed [16]. It has been also recognized that string theory or M-theory may lead to brane-world models in which one of the extra dimensions can even be non-compact [17,18]. In all these models, the fourdimensional world is a brane, on which the matter fields live, while gravity, and in some interesting cases the gauge and the Higgs fields, propagate also in the transverse extra dimensions of the bulk space.
In the majority of the cases studied, in an attempt to build realistic models, the bulk is a five-dimensional space [19]. In these models the corresponding backgrounds may be of Minkowski or anti-de-Sitter type. The effects of the above considerations in specific GUT models have been also considered. The assumed background in these models is of Minkowski type and questions regarding the unification and supersymmetry breaking scales have also been addressed. In this direction, assuming that the fifth dimension is very large, of the TeV scale, non-supersymmetric extensions of the SM even without the need of unification have been considered [20,21,22,23,24,25,26,27]. On the other hand, models embedding the SM in an anti-de-Sitter five-dimensional space have been also discussed [28,29].
In view of the aforementioned developments the study of the five-dimensional supergravities has been revived [30,31,32,33,34,35]. This is quite natural since, after all, gravity is in the centre of all these attempts and it is legitimate to assume that we have to treat the fifth dimension before going to the 'flat limit'. In the framework of five-dimensional supergravity no specific model based on a particular gauge group has been studied so far.
In this paper we consider five-dimensional supergravity with the minimal unified gauge group, SU (5). The gauge symmetry enters with a minimal number of multiplets following the construction of [31]. We study the emerging potential and discuss the Minkowski or anti-de-Sitter character of the classical vacua. We comment on the fate of the gauge symmetry and supersymmetry of these vacua, ignoring at this stage the inclusion of matter fields, which do not play any role in this consideration.

Setting the model
We consider a five-dimensional gauged supergravity model with SU(5) gauge symmetry. The field content of the model is [30] where singlet, labelled 's', which is introduced for reasons that are explained later. The field content is completed by ten tensor multiplets, M = 1, . . . , 10, transforming as 5 +5 under SU (5). The spinor and the scalar fields included in the vector and tensor multiplets are collectively denoted by λ iã and ϕx, respectively. In particular the scalar fields of the model are ϕx = {ϕ a , ϕ s , ϕ M }. The indicesã andx are the flat and curved indices, respectively, of theñ-dimensional manifold M which is parametrized by the scalar fields. This manifold is embedded in an (ñ + 1)-dimensional space and is determined by the cubic constraint whereĨ,J,K = 0,x. hĨ are functions of the scalar fields defining the embedding of the manifold M. CĨJK are constants symmetric in the three indices.
For the model at hand we choose hx = ϕx, where ϕx are arbitrary and h 0 is to be determined by the constraint given by equation (2). This choice is convenient since in this case the constraint becomes manifestly SU (5) invariant. Taking the constants CĨJK as explained in [31] the constraint reads where Φ = ϕ a K a , a = (1, . . . , 24), is in the adjoint representation of SU (5), in the form of a 5 × 5 matrix. K a denote the SU(5) generators normalized as Tr(K a K b ) = 1 2 δ ab . χ are scalars in the fundamental representation of SU (5). The fields ϕ M introduced previously are just the real and imaginary parts of the fields χ. In particular with The bosonic part of the Lagrangian which is SU(5) × U (1) R symmetric is given by [31] The corresponding gauge fields are A a µ and where V 0 and V S are constants. The option in which only SU(5) is gauged is implemented by putting g R = 0. In this case the 's' multiplet is redundant. All scalars are neutral under U (1) R while under SU(5) we have an adjoint, a (5 +5) and a singlet. The covariant derivatives appearing in the Lagrangian are both general coordinate and gauge covariant. The fields HĨ µν collectively describe the field strengths of the vector fields and the self-dual tensor fields The constants f K IJ denote the structure constants of the gauge group. f c ab are the usual SU(5) structure constants and f K IJ = 0 if one of the indices is 0 or s, thus satisfying the condition V I f I JK = 0. g and g R are the SU(5) and U (1) R gauge coupling constants, respectively.

1.4
The tensor Ω MN in the Lagrangian (6) is given by the matrix As far as the tensors appearing in the kinetic terms are concerned,åĨJ is the restriction of the metric of the (ñ + 1)-dimensional space on theñ-dimensional manifold of the scalar fields and is given byåĨJ where hĨ = CĨJKhJ hK =åĨJ hJ .
In equation (6) gxỹ is the metric of theñ-dimensional manifold M given by where hĨx = − 3 2 hĨ,x. Note also that the following relations hold hĨhĨ = 1, with hĨx = 3 2 hĨ,x. Supersymmetry invariance requires the existence of additional potential terms in the Lagrangian. In particular, the U (1) R gauging gives rise to the potential g 2 with Furthermore the existence of tensor multiplets leads to the appearance of g 2 P where P = 2WãWã (15) with Wã given by Ω MN is the inverse of Ω MN given in equation (8). The conversion of flat indices (ã) to curved ones (x), and vice versa, is implemented by the use of fã x , the vielbein of the manifold M. For the specific model, the quantities hĨ are given by The metric can be written as (see equation (9)) In equation (19) only the non-vanishing components of bĨJ are shown. Note that, at the point h 0 = 1, ϕx = 0, we haveåĨJ = δĨJ as required by the consistency of the constraint (3).

The potential
The part of the potential due to the scalar fields which are non-singlets under the gauge group and do not belong to the gauge multiplets is (see equation (15)) Using the complex notation for the fields in the (5 +5) representation we find that Thus the non-Abelian part of the potential receives the simple form where the vector Mx is given by From equation (22) we see that in order to study the potential we need the inverse of the metric gxỹ. With the adopted form of the constraint, the metric takes the form where The inverse of the metric is then found to be where Fxỹ is the inverse of Fxỹ in equation (25). In view of the above relations the non-Abelian part of the potential can be written as 1.6 The potential stemming from the U (1) R gauging consists of two terms. The first is negative definite and the second is positive With our choice of coordinates hỹ x = − 3 2 δỹ x , and using the relation (12), we find that h 0 where the corresponding vectors are given by Using the form of gxỹ found earlier the positive term of the Abelian part of the potential is given by The full scalar potential is given by and it is rather involved to deal with analytically. Therefore we will proceed numerically in order to study its vacuum structure.

Comments on the vacua
The basic features of our analysis of the scalar potential may be summarized as follows.
(i) If the U (1) R symmetry is not gauged, then we have a positive definite potential. This potential has flat directions along χ A = 0 which preserve supersymmetry. Thus we have a class of supersymmetric Minkowski vacua in which the gauge group may be broken only by the scalar fields in the adjoint representation. The degeneracy of the vacua may be lifted by radiative corrections. If this is the case the gauge symmetry breaking occurs in the bulk.
In the opposite case we have a supersymmetric SU(5) invariant vacuum [19]. (ii) The U (1) R gauging induces, as already mentioned, a negative contribution to the potential for supersymmetry to be preserved. This contribution alters the situation drastically as we shall see in the sequel.
In what follows we will study the symmetry breaking patterns SU(3) × SU(2) × U (1), SU(4)×U (1) and SU(2)×SU(2)×U (1) occurring since the fields in the adjoint representation may develop non-vanishing vacuum expectation values (VEVs) along the appropriate directions. For the fields χ A , χĀ we take only χ 5 = χ5 to be non-zero, to avoid charge violation.

1.7
As a first step we consider the model without the multiplet (A s µ , λ si , ϕ s ). This multiplet, unlike the rest, is not necessary in order to construct a locally supersymmetric Lagrangian with SU(5) × U (1) R gauge symmetry in five dimensions. In this case we find that the potential has a local maximum at the SU(5) symmetric point, but it develops instability in the directions χ A = χĀ = 0. Along these directions the positive contribution of P disappears and we are left only with P (R) . In P (R) the negative term −(h 0 ) 2 , where h 0 is the real root of the constraint which is continuously connected to unit, dominates the positive term. The instability is caused by the fact that this root continues to grow as ϕ a increases. Although this sort of instability is not peculiar for these models, see for example [34], it is an unpleasant feature for a realistic model.
In an attempt to lift the instability in a minimal way we proceed to the inclusion of the multiplet {A s µ , λ is , ϕ s } which is scalar under SU(5) as we have said. Let us remark at this point that the lowest dimensionality of the scalar manifold arising from string theory is 35 (see [4] (vol 2, p 309)). This is the case in the model under consideration after the introduction of the singlet field ϕ s . The introduction of an extra SU(5) inert multiplet allows for a combination for the U (1) R gauging, and correspondingly modifies the negative term of P (R) to with the appropriate change in the positive term. In this case it turns out that we still have the instability in the directions χ A = χĀ = 0. In fact for large values of h 0 , ϕ s the negative term (35) dominates over the corresponding positive term. Then the SU(5)-symmetric point remains either a local minimum or turns to a saddle point.
From the study of the potential we see that with this minimal number of multiplets it is impossible to get rid of the instability if the range of ϕ s is infinite. A possible way to remedy the situation is by restricting the field ϕ s to take values in a finite range. For instance, solitonic solutions of finite variation (kinks) for the scalar fields of five-dimensional supergravity have long been known [36]. Recently, explicit examples of such solutions have been worked out for the case of a supersymmetric domain wall world embedding in the context of five-dimensional gauged supergravity [32,37]. Furthermore, when five-dimensional supergravity is obtained from compactification of eleven-dimensional supergravity on a smooth Calabi-Yau manifold such neutral scalars emerge naturally [32,38]. On these grounds it is legitimate and well justified to allow the field ϕ s to vary in a finite range.
From the study of the potential we find that for each value of ϕ s and V s there is a maximum value for V 0 for which stability of the potential is guaranteed. This means that there are nontrivial combinations of A 0 µ and A s µ for the U (1) R gauging that lead to a stable potential. For instance, if we arbitrarily fix the values of ϕ s and V s and choose V 0 = 0, the absolute minimum of the potential occurs when all non-singlet scalars, under SU(5), get vanishing VEVs. Therefore the vacuum is at the SU(5) symmetric point. The difference now is that we have a negative cosmological constant given by Varying V 0 we find that SU(5) breaking minima develop due to the contribution of h 0 in equation (35) which is determined by the constraint. The SU(5) symmetric point turns out to be either a local maximum or a local minimum, but in the latter case it is not the absolute minimum.  We have found that after some critical value of V 0 the potential becomes unstable. We point out that, at the level of five-dimensional supergravity, the relative values of V s and V 0 are arbitrary. Therefore it is a matter of the more fundamental theory to determine their values.
In figures 1 and 2 we display the potential for particular values of ϕ s , V s and V 0 . Since we are interested in the directions χ A = 0, only the Abelian part is plotted. In direction. The corresponding minimum is slightly deeper from the phenomenologically acceptable SU(3) × SU(2) × U (1) vacuum. Note however that the SU(4) × U (1) direction may be excluded from the geometry of the manifold since in this direction, unlike the other cases, the metric has singularities. Certainly we do not claim that the model selects in this case the right vacuum naturally but the fact that the gauge symmetry is broken as a consequence of the particular U (1) R gauging is by itself very interesting.
As far as the supersymmetric nature of the vacua is concerned, it should be remembered that ϕx = 0 and λ ix = 0 implies for the supersymmetry transformations that δλ ix = 0. This in turn implies that, in order for the supersymmetry to be preserved, we must have Pã = Wã = 0 when the fields are on their VEVs. Ignoring zero modes of the fünfbein the above relations imply that Mx = Nx = 0. However, in the model at hand, the vacua which break the gauge symmetry have Pã = 0 and therefore supersymmetry is broken. It should be noted that the VEVs of the scalar fields enter the supersymmetry transformations of the spinors through the combination gWã i + 1 Then supersymmetry for the vacua implies that So, assuming that 2 = 1 , the above relations read If the VEVs and the coupling constants are such that either of equations (37) is satisfied, then we have broken supersymmetry in the bulk space but N = 1 supersymmetry in the fourdimensional space. Although very interesting, this possibility is ruled out on the grounds that a non-vanishing value of Wã = 0 is required since Pã = 0 too. This adds a positive contribution to the non-Abelian part of the potential making these d = 4 supersymmetric vacua not the absolute minima of the theory. Thus, in general, only the SU(5) symmetric point is supersymmetric. In any other case we have simultaneous breaking of both supersymmetry and the gauge symmetry. The model at hand is by itself an effective one so it may not be inconceivable that ϕ s does not vary within a finite range but the dynamics of the underlying theory drives ϕ s to a fixed value. In this case the instability is also lifted. A consequence of this is that supersymmetry is completely broken. This is due to the fact that the field ϕ s belongs to the same multiplet with the vector field A s µ which participates in the U (1) R gauging. Therefore, the mechanism that fixes ϕ s to a constant value-in fact sets this to be a moduli field-is intimately connected with the supersymmetry breaking mechanism.
In order to reach a firm conclusion which one of the aforementioned possibilities for stabilizing the potential can be realized and exploring the supersymmetry properties of the vacua, a more detailed analysis of the scalar field manifold is needed. Note that the manifold M describing this model is neither maximally symmetric nor homogeneous [31] lying therefore outside the cases that have been extensively studied and classified so far [30,31,39]. Besides this geometrical analysis an explicit construction of the model from the eleven-dimensional supergravity is necessary in order to clarify which one of the particular vacua is favoured.

Conclusions
We have studied the classical vacua of a five-dimensional SU(5) supergravity model with the minimal field content and no matter fields. We have found that: (1) Without U (1) R gauging, the potential has flat directions determined by the vanishing VEVs of the fields in the 5 +5 representation of SU (5). Along these directions supersymmetry is preserved. Certainly this model has to be completed with the matter spectrum. The effective four-dimensional model will arise either after compactification of the fifth dimension or by considering the corresponding action on a three-dimensional brane. In any case, the five-dimensional vacuum affects the four-dimensional theory. The fact that, at the classical level, the gauge symmetry and the supersymmetry breaking do not occur at the five-dimensional bulk means that such models are characterized by a large scale of effectiveness (energy desert [40]). The possibility that the degeneracy is lifted by five-dimensional radiative corrections supports models that do not exhibit unification in four dimensions. (2) The U (1) R gauging creates stability problems. In order to overcome the instability we introduce a vector multiplet neutral under the gauge group. The mechanisms proposed are either the restriction of the variation of the corresponding scalar field, ϕ s , in a finite range or fixing this field to a constant value. The former case may be realized by kink solutions existing in the context of five-dimensional supergravity, while the latter is an issue related to the dynamics of the underlying fundamental theory. Under these assumptions we have found that the potential becomes stable in a certain range of the parameters V 0 and V s , which determine the mixing of the graviphoton and the vector field of the extra multiplet which plays the role of the U (1) R gauge field. The vacua in this case are of the anti-de-Sitter type. As far as the fate of the gauge symmetry is concerned the choice V 0 = 0 yields an SU(5) invariant minimum. In this case supersymmetry is unbroken if ϕ s varies in a finite range but is broken if ϕ s is fixed to a constant value. When V 0 = 0, SU(5) breaking minima appear. In all the minima where the gauge symmetry is broken supersymmetry is also broken. As is already quoted the five-dimensional vacuum determines the scale of effectiveness of the corresponding brane-world model. For example, the SU(5) breaking vacua are compatible with the scenaria with low-energy scale brane description. The non-supersymmetric SU(5) vacuum may be compatible with unification with low string scale [41].
Thus we see that the scenaria for physics beyond the SM may in principle be incorporated in a unified five-dimensional supergravity content. Certainly the properties of the scalar field manifold, the mechanism for stabilization of the potential in the case of anti-de-Sitter space and the details of including matter on the brane describing the four-dimensional space require further investigation. Work towards this direction is in progress.