On the Stability of D7 - anti-D7 Probes in Near-conformal Backgrounds

We investigate the perturbative stability of a nonsupersymmetric D7 - anti-D7 brane embedding in a particular class of type IIB backgrounds. These backgrounds are the gravitational duals of certain strongly-coupled gauge theories, that exhibit a nearly conformal regime (known as a walking regime). Previous studies in the literature have led to conflicting results as to whether the spectrum of fluctuations around the flavor D7 - anti-D7 embedding has a tachyonic mode or not. Here we reconsider the problem with a new analytical method and recover the previously obtained numerical results. We also point out that the earlier treatments relied on a coordinate system, in which it was not possible to take into account fluctuations of the point of confluence of the D7 - anti-D7 branes. Using an improved coordinate system, we confirm the presence of a normalizable tachyonic mode in this model, in agreement with the numerical calculations. Finally, we comment on the possibility of turning on worldvolume flux to stabilize the probe branes configuration.


Introduction
Theories that are nearly conformal for a large range of energy scales, such as walking gauge theories [1], could exhibit non trivial dynamics, but are notoriously hard to analyze due to their strong coupling. Gauge/gravity duality [2,3,4], which maps quantum dynamics of strongly coupled gauge theories to classical solutions of gravitational theories in higher dimensions, is a promising technique to analyze such models. Using this method, we studied recently a model of a walking gauge theory, that may be relevant for phenomenological applications related to the LHC. More specifically, in [5] one of us introduced U-shaped stacks of D7-D7 branes to incorporate chiral symmetry breaking, a la Sakai-Sugimoto [6], in the gravity background of [7]. Subsequently, we investigated the spectrum of vector [8] and scalar mesons [9] in this model. The scalar boson spectrum was obtained from studying the fluctuations of the D7-D7 branes. Using an approximate linearized version of the relevant equations of motion, we obtained a spectrum of fluctuations with all eigenvalues E = m 2 positive, corresponding to a stable embedding of the branes. It turns out, however, that this consideration did not capture the full picture.
The position of the D7-D7 embedding in the transverse space can be described by two angles θ and ϕ, which are functions of a radial variable z running over both the D7 and D7 branes. The equations of motion for the scalar mesons, arising from fluctuations around that embedding, are second-order coupled differential equations for the deviations inappropriate choice of coordinate system.
In Section 4 we reconsider the δθ spectrum with the new method developed here, that is based on the investigation of zero mass modes of the equations of motion. We find instabilities in this sector both for Neumann and Dirichlet boundary conditions at z = 0.
However, unlike the instability in the δϕ mode, these instabilities satisfy the regularity condition. Nevertheless, they are not expected to be present in phenomenologically relevant models. The reason is that, in the parameter range required to have a slowly running gauge coupling, these instabilities occur when the cutoff z Λ is chosen to be extremely large.
Such a choice of z Λ , however, is incompatible with the phenomenological constraints in this kind of model [8]. 7 In Section 5 we argue that the field ϕ used so far, here as well as in the previous literature, does not capture the full scalar spectrum. The reason is that it does not describe a transverse direction everywhere along the probe D7-D7 embedding. Hence, it may miss information about the scalar meson spectrum, which is supposed to arise from the transverse fluctuations. We then introduce appropriate new coordinates and show that there is a physical tachyonic mode, signaling pertubative instability. 8 We point out that the model could be stabilized by turning on worldvolume flux, in the vein of [12], but we leave this interesting question for the future.
In Appendix A we discuss in more detail the singular solution in the δϕ sector. The singularity occurs because the transformation between the Schrödinger equation and the original equation of motion is singular at z = 0. This leads to δϕ having a pole at z = 0.
We recall two separate arguments for requiring the physical modes to be regular.
In Appendix B we prove a set of theorems, that show the relationship between zero energy solutions and negative energy ones in a Schrödinger equation with a cutoff. These are needed to obtain the results of Sections 3 and 4. We should note that the new method for studying questions of (in)stability, that arises from this appendix, is a rather powerful tool for investigating nonsupersymmetric brane embeddings in nontrivial backgrounds. 7 Note that these constraints were derived from the vector meson spectrum, which does not contain a tachyon. One may expect the inclusion of background flux for the worldvolume gauge field, needed to remove the scalar spectrum tachyon, to preserve the previous conclusions regarding the phenomenological constraints. However, it should be kept in mind that this issue merits a separate investigation. 8 In fact, up to the Jacobian of the relevant transformation, that mode agrees with the result of [11].
2 Set-up and first look at stability issue The system of interest for us arises from considering probe D7 branes embedded into the background of [7]. The latter background is an N = 1 supersymmetric solution of type IIB, sourced by a certain stack of D5 branes. The metric in that background depends on two parameters α and c. The solution of [7] is given as an expansion in the parameter 1 c < < 1 and, to leading order, has the form: 2) ω 1 = cos ψdθ + sin ψ sinθdφ , ω 1 = dθ , ω 2 = − sin ψdθ + cos ψ sinθdφ , ω 2 = sin θdϕ , and There is also nontrivial RR 3-form flux, which is not relevant for the following. 9 This background is the gravitational dual to a walking gauge theory. In the walking regime, the above metric simplifies to [9,10]: where we have introduced the notation β ≡ sin 3 α < < 1 . (2.6) A realistic model of dynamical electroweak symmetry breaking would need to encode, at energies (radial distances) above the walking region, a larger symmetry group known as extended technicolor [16]. Modifying the above solution, to take this into account, 9 Note also that, to leading order in 1/c, the string dilaton is constant in this solution.
is still an open problem, although there has been recent progress in that direction [17].
So, for practical reasons, we have introduced a cut-off ρ Λ [8], which is the physical scale corresponding to the symmetry breaking of that larger gauge group to the smaller group of the walking regime. Our model, thus, uses the background (2.5) with ρ ≤ ρ Λ .
To introduce flavor degrees of freedom and study dynamical chiral symmetry breaking in the above set-up, one can show that there is a U-shaped embedding, similar to [6,14], of D7 − anti-D7 probe branes in (2.5); see [5]. Let us denote the radial position of the tip of that embedding by ρ 0 . 10 In [9] we introduced for convenience a new radial variable by: The conceptual difference between ρ and z is that ρ is a spacetime radial coordinate, whereas z runs only over the worldvolume of the D7-D7 embedding. The shape of the embedding in the two transverse directions is described by the fields θ(z) and ϕ(z), which parametrize the position of a worldvolume point in the transverse (θ, ϕ) two-sphere. The scalar mesons in this model arise from space-time dependent fluctuations around the classical shape of the embedding. Namely, they are described by the fields θ(z, x µ ) = θ cl (z) + δθ(z, x µ ) and ϕ(z, x µ ) = ϕ cl (z) + δϕ(z, x µ ), where x µ are the four-dimensional coordinates in (2.5). To find the Lagrangian for these fields, one begins with the standard DBI action that describes the D7 branes, i.e.
It is important to keep in mind that the theory we consider has a cutoff at a finite value of z, denoted by z Λ , which corresponds to the physical cut-off ρ Λ mentioned above. 10 Note that the shape of the embedding around the tip is smooth. 11 Note that, in (2.14) of [9], we have substituted with its background value the sin 2 θ multiplier in front of the ϕ 2 µ term in the last bracket. This is because it leads to higher (than second) order in the fluctuations; see the discussion in Appendix A of [9]. Here we retain this multiplier for completeness.
Furthermore, though the range of z is −z Λ ≤ z ≤ z Λ , it is sufficient to find solutions in the range 0 ≤ z ≤ z Λ , due to the symmetric or anti-symmetric boundary conditions imposed at z = 0.
One can easily verify [9,5] that a simple classical solution of the Euler-Lagrange equations for θ and ϕ is given by θ cl = π/2 and (2.10) To investigate the perturbative stability of this solution, one needs to study small δθ and δϕ fluctuations around this configuration. It is easy to see that the two kinds of fluctuations decouple to second order in the expansion of the Lagrangian.
Let us first consider the fluctuations of ϕ. To second order, the effective Lagrangian has the form: where for convenience we have suppressed the dependence of δϕ on the space-time coordinates x µ and, also, we have introduced the notation ϕ = ∂ z ϕ. For plane wave solutions with mass m, we can rewrite the Lagrangian as: where we have denoted M 2 ≡ c γ m 2 / 12 . The equation of motion, following from that Lagrangian, is: To be able to estimate analytically the spectrum at leading order, in [9] we rephrased the problem of solving (2.13) into a quantum mechanical problem for a Hamiltonian where see Section 4 there. The key use of this was that the equation for the eigenfunctions χ and eigenvalues M 2 of H 0 can be easily solved analytically exactly.
The solutions given in (4.14) of [9] are: where J 0 is the Bessel function. Having this result, one can use quantum mechanical perturbation theory to find the corrections that are due to the Hamiltonian ∆H 0 . The outcome is written down in [9]. That it does not contain any tachyon mode is due to the fact that the leading spectrum, which corresponds to (2.17), does not have any negative mass-squared modes. The correction due to ∆H 0 is subleading in γ, compared to the eigenvalues M 2 of H 0 , and therefore cannot change the positivity or negativity of the total answer for the mass-squared.
Note, however, that the solution (2.17) already assumes M 2 > 0. So to verify that there are no tachyonic modes in the above consideration, let us consider in more detail the case M 2 < 0. In such a case, the solution to (2.16) is the following combination of modified Bessel functions: where c 1 and c 2 are arbitrary constants. However, the physical solutions in this model have to satisfy the conditions that they are regular 12 and vanish at z = z Λ . 13 Now, it easy to realize that these conditions rule out any solution of the form (2.18). More precisely, K 0 is singular at the origin z = 0, while I 0 is monotonic and everywhere positive. This led us in [9] to the conclusion that there is no negative mass-squared mode in the spectrum.
However, as we explain in a later section, the choice of coordinates used so far is not the most appropriate one. As a result, it misses the lowest fluctuation mode, which turns out to be a tachyon. Before turning to that, it will be useful and illuminating for the future to reconsider the numerical work of [11], the first to indicate an instability in this model, via analytical means.
To make the comparison to [11] more transparent, we will now study the Schrödinger form of the equation of motion. As pointed out in [9], equation (2.13) can be transformed to the Schrödinger form via the field redefinition (2.20) 12 For more on this condition, see Appendix A. 13 This is precisely why the solution (2.17) does not contain the Y 0 Bessel function.
In the next section we turn to studying equation (2.19). We will find by analytical means the tachyonic mode of [11]. However, the latter will not translate to a physical mode, as it corresponds to a singular solution of (2.13), despite being a regular solution of (2.19). The underlying reason for the physical inequivalence of the two formulations, i.e. equations (2.13) and (2.19), is that the transformation (2.20) is singular at z = 0. The point z = 0 is of crucial importance, as this is where one must impose the boundary conditions determining the spectrum in the Schrödinger picture.
Nevertheless, we will show in Section 5 that there is, after all, a tachyonic mode in this model. However, to see that one needs to use an improved choice of coordinates that separates properly the transverse and worldvolume directions with respect to the U-shaped probe D7-D7 embedding.

Schrödinger equation and stability
Let us rewrite equation (2.19) as: This is in complete agreement with the Schrödinger equation and potential studied in [11]. The latter work used numerical methods to find the spectrum.
We would like to investigate (3.1) with analytical means. In order to do that, we will use a set of theorems recalled in Appendix B. The main point is that, to find whether or not there is a tachyonic mode, one only needs to look for zero mass solutions. If there is such a solution and it vanishes at some point z c > 0, then there is a tachyonic mode for If there is no zero mass solution or the M = 0 solution does not vanish anywhere, then the spectrum does not contain any tachyonic modes. The precise derivation of the relevant set of quantum mechanical theorems is given in Appendix B. But the conceptual picture behind them is the following. The values of the mass levels, m 2 n , depend on the value of the cutoff z Λ . More precisely, by decreasing z Λ one increases m 2 n monotonically. Furthermore, at low enough z Λ , all modes have positive mass-squareds m 2 n . Therefore, if by increasing sufficiently z Λ , we can find a point z Λ = z c , such that for the lowest level m 2 0 (z c ) = 0, then when z Λ > z c we will have m 2 0 (z Λ ) < 0 and thus a tachyon. Now, recall that a boundary condition for the physical solutions of (3.1) is that they vanish at z Λ .
Hence, if there is a zero mass solution vanishing at a certain point z c , then for z Λ > z c the spectrum contains a tachyonic mode.
In view of the above, we now turn to studying the zero mass solutions of (3.1). First, note that it has complete sets of solutions with both Neumann and Dirichlet boundary conditions at z = 0. At the same time, every solution must satisfy the boundary condition where c D and c N are integration constants. Clearly, for Neumann boundary condition at This solution diverges at z = 0 and the key reason for that is the singularity of the transformation (2.20) at the origin. As a result, the solution (3.4) would be ruled out by the regularity condition 14 that needs to be imposed, just as was the case in the language of the previous section. However, as already alluded to above, the coordinates used so far are not the best suited for properly capturing all the information about the scalar spectrum. We will discuss a better choice in Section 5 and show that there is a tachyonic mode after all.
As a final comment, it is easy to show by series expansion that the massive solutions of (2.13) with Neumann boundary condition in the Schrödinger form also have a pole at z = 0, just like the massless one.
14 The regularity condition is well-known in the literature. Nevertheless, for more completeness, in Appendix A we give some arguments as to why singular solutions are inadmissible.

The δθ spectrum revisited
In this section, we will reconsider the fluctuations of θ with the new method developed here. As we have shown in [9], the equation of motion for δθ, that follows from the Lagrangian (2.9), can be converted to Schrödinger form via the field redefinition The result is: where the odd Dirichlet solution and the even Neumann solution are combinations of Legendre functions, namely: We should note that, for illustrative purposes, we have used values of γ that are (significantly) larger than the phenomenologically acceptable range (see [8]). This is because for smaller values of γ the zeros of Θ D and Θ N occur at extremely large z, which makes plotting very difficult.
Nevertheless, the asymptotic form (at large z) of (4.4) makes the location of the zeros of Θ D and Θ N rather clear. Namely, one can show that at large z:  The location of the zeros, in the limit of small γ, can then be read off from (4.5). In particular, the smallest zeros correspond to the following values of z:  precisely, the latter means the requirement that the S-parameter in this model is consistent with observation. Hence, our phenomenological fits in [8] are in the region of stability.
Note that the presence of an instability in the θ sector, for a large enough cutoff, was found in [11] with an entirely different method; see the discussion of Model 1 in their Appendix. Our estimate (4.6) for the critical cut-off, above which it sets in, matches perfectly with the one given in their equation (44).
Finally, let us comment on why the theoretical instability in the δθ sector was not discovered in [9]. The reason is that z N c and z D c are non-perturbative functions of γ and, in particular, z N,D c → ∞ when γ → 0. In [9], we treated the correction term in (4.2), namely the one proportional to γ, as a small perturbation. Therefore, results non-perturbative in γ were beyond reach for the methods used there.

Cartesian coordinates and instability
In the previous sections we described the U-shaped embedding of the probe branes as a function ϕ(z), where the worldvolume radial coordinate z is related to the spacetime one ρ via z = ± √ e 4(ρ−ρ 0 ) − 1; see [9]. However, the 10d spacetime coordinate ϕ is not always transverse to the U-shaped embedding. Therefore, it may not capture the full information about the scalar field spectrum, which arises from fluctuations of the transverse coordinates. Indeed, close to the tip, which is given by ρ = ρ 0 , the radial coordinate ρ is a better approximation to a transverse coordinate, whereas far away from the tip the angular coordinate ϕ plays such a role. Hence, to find an appropriate choice, such that one coordinate remains transverse and the other one remains longitudinal everywhere along the embedding, we need to perform a change of variables that mixes ϕ and ρ.
It turns out that the appropriate new coordinates can be defined by: The reason is the following. The classical embedding solution, as a relation between ϕ and ρ, is given by [5]: Rewriting this as: In that regard, note that it obviously runs from −∞ to +∞ and, furthermore, when ρ = ρ 0 we have ϕ = 0, which impliesẑ = 0. Finally, let us also point out that (5.1) implies the "Cartesian" relation: 5) which is rather similar to the situation in both [14] and [6], after they introduced new coordinates to study the scalar spectrum. The general solution of this equation is The solution with Neumann boundary condition atẑ = 0 has a zero at finiteẑ =ẑ c , implying, according to the set of theorems in Appendix B, that the classical solution is unstable if the cutoffẑ Λ >ẑ c 1.5 × e 2 ρ 0 .
One can make the instability more explicit by diagonalizing the quadratic effective Hamiltonian for fluctuations. Introducing δY = δy (e 4 ρ 0 +ẑ 2 ) in (5.6) we obtain for the quadratic Hamiltonian The negative signs of the last two terms in (5.9) are in contrast to the similar expression for the Sakai-Sugimoto model [6], in which all terms have positive signs. This difference has a crucial implication. Namely, the Hamiltonian in our case must have a negative eigenvalue, which signifies instability. To realize this, note that the Hamitonian (5.9) takes a negative value at the trial function δY = const.
Finally, let us discuss the broader issue of how much the entire mass spectrum, obtained from working in the Cartesian coordinate system introduced in this section, can differ from the spectrum obtained by using the old coordinate system. To answer this question, notice the following. Upon substituting in (5.6) the ansatz δy(ẑ) = (1 + where ζ =ẑe −2ρ 0 , one can verify that the resulting equation of motion becomes identical to (3.1), with Ψ replaced by δu and z by ζ. Therefore, the two spectra are identical, modulo some modes that may be present in one of them but missing in the other due to different boundary and/or normalizability conditions in the two coordinate systems. We have already seen that the tachyon is such a mode, present in the Cartesian coordinate system but missing in the old one. It is worth noting that the old coordinate system may, in principle, be missing other modes as well, since it does not split properly between transverse and longitudinal directions to the probe brane's worldvolume.

Conclusions
We reconsidered the problem of stability of the D7-D7 embeddings studied in [9,11]. We developed a new analytical method, that allowed us to reproduce the numerical results of [11]. The method relies on the connection between tachyons and zero mass modes in a Schrödinger equation with a bounded potential and a finite range for the variable.
Furthermore, we pointed out that, although the equations of motion studied in [9] and in [11] are mathematically equivalent, they do not capture the same physical information in the ϕ sector. The reason is that the transformation, relating them to each other, is showed that the scalar spectrum does contain a tachyonic mode. Up to the Jacobian of the relevant coordinate transformation, that mode agrees with the tachyon of [11]. It would be very interesting to explore stabilizing the model by turning on worldvolume flux as in [12,13]. We hope to come back to this question soon.
It is also interesting to note that, with our new analytical method, we confirmed the presence of a θ sector instability, in agreement with [11]. More precisely, we showed that the spectrum of δθ fluctuations contains a tachyonic mode, when the cutoff of the model exceeds a certain value. This value is extremely large, being given by an exponential of an inverse power of the small parameter γ. Thus, it is well beyond what is allowed by the requirement for a phenomenologically acceptable value of the S-parameter [8], when this model is viewed as realizing dynamical electroweak symmetry breaking. 16 Nevertheless, this is an instability that exists in the theoretical model, studied here, of a flavored strongly coupled gauge theory with two dynamical scales, which are separated by a nearly conformal region. It is conceivable that this gauge theory might have other applications in the future 17 , in which the new instability could play a role. 16 A clear caveat is that one needs to investigate first whether the presence of worldvolume flux modifies the phenomenological constraint or not. 17 For example, an intriguing new application of the walking background of [7] is within the context of the recently proposed model of slow-walking Cosmological Inflation [15].

L.
A. is also supported in part by funding from NSERC. The research of R.W. is supported by DOE grant FG02-84-ER40153.

A On singular solutions for δϕ
In this Appendix we review a couple of arguments as to why one needs to impose the regularity condition for solutions of the equations of motion in our context. It should be kept in mind that the appendix relies entirely on the coordinate system used by [9,11], in which the fluctuating field does not have a transverse component everywhere along the D7-D7 brane embedding. Therefore, one is not guaranteed to capture the full spectrum.
In Section 5 of the main text, we will address this issue by introducing new coordinates, in which the fluctuating field is transverse everywhere along the embedding. In the new coordinates, we will find a tachyonic mode given, up to the relevant Jacobian factor, by the same expression as the one discussed here and in [11].
As we have seen in Section 3, a singular solution of (2.13) arises from solving (3.1) with Neumann boundary condition at z = 0. The first argument against accepting such a solution as part of the physical spectrum is the requirement for a finite action for any perturbative mode. To see how the latter is violated by the singular solutions, let us consider the kinetic term for δϕ fluctuations in the action: 18 One should keep in mind though, that the consideration here refers to the coordinate system used in [9,11]. As already mentioned above, in Section 5 we will introduce a more suitable coordinate system that will lead to a different conclusion regarding the tachyon mode.
Another argument for requiring regular solutions is the following. The quadratic action, on which the analysis of perturbative (in)stabilities relies, should not break down as the leading approximation to the full action. To see that in our case there is such a break down, let us consider in some detail the power series expansion of the quadratic action in a neighborhood of the point z = 0. First, note that the terms containing δϕ µ in (2.9) have a pre-factor z 2 . Therefore, when calculating the expansion coefficients of δϕ n near z = 0, we may neglect δϕ µ altogether. For simplicity, we can also neglect contributions from δθ, which leaves us with the following approximate action: valid in a neighborhood of z = 0. Now, expanding L 0 z in δϕ (z), we arrive at the following schematic expression for the action near z = 0: This power series is convergent only if δϕ (z) < ϕ cl (z) γ / 3 , where the last relation follows from (2.10). The convergence condition is clearly violated for all solutions that are singular at z = 0. In particular, this means that it is violated for all solutions of the Schrödinger form equation, which satisfy a Neumann boundary condition at z = 0. As we have already seen, the tachyonic mode in the δϕ spectrum belongs exactly to this class of solutions. In other words, it is outside the range of validity of the action, whose equations of motion it solves. Thus, it is not clear whether it is a legitimate solution of the physical problem at hand.
Note that, although in our case the first argument we gave above is, clearly, stronger than the second one, the second argument is worth mentioning for the following complete- Let us consider the Schrödinger equation with a potential V (z) that is a real analytic function of z for all z ≥ 0. In fact, it will be enough to require the considerably weaker condition of piecewise analyticity and continuity of V (z), together with the constraint |r V (r)| < ∞ at z = 0. We will take the variable z to vary within the interval 0 ≤ z ≤ z Λ . Also, we will impose the boundary condition that at the cutoff z Λ all eigenfunctions vanish, i.e. ψ n (z Λ ) = 0. Upon imposing appropriate boundary conditions at z = 0 (the form of which will be discussed below), equation (B.1) has a discrete spectrum with infinitely many eigenvalues. We will investigate the dependence of those eigenvalues on z Λ . In particular, we will address the question of when there are eigenfunctions with negative eigenvalues E < 0, whose presence would indicate an instability.
In what follows, we will present a series of theorems, which allow us to reach the conclusion that the instability for a cutoff z Λ is related to the existence of a zero mass solution for a smaller cutoff z λ < z Λ . We do not claim that all of these theorems are original, but we list them for completeness. Theorem II: Consider two potentials V 0 (z) and V 1 (z). Suppose that V 1 (z) ≥ V 0 (z) for every z and that, at least in one finite interval, we have V 1 (z) > V 0 (z). Let the ground state energy of equation be E 0 . Then, the inequality E 1 > E 0 holds.
Indeed, this statement is true for the following reason. If we denote the ground state eigenfunctions by ψ 1 and ψ 0 respectively, then we have: (B.5) Let us consider the expression Hence E 0 < E 2 < E 1 , because ψ 1 (z) must be a mixture of the ground state and excited states of the equation with potential V 0 . Therefore, the ground state eigenvalue is lowered by decreasing the potential.
Theorem III: The ground state energy for a potential, bounded from below, is positive at a sufficiently small value of the cutoff z Λ .
This theorem can be proven as follows. First, let us denote by −N the lower bound of the potential V . Now, according to Theorem II, the ground state energy for V is larger than that for the potentialṼ Clearly, in both cases the ground state energy is larger than 0 at sufficiently small z Λ .
Of course, this theorem is essentially a consequence of the Heisenberg uncertainty relation in free space. Note also that the same statement can be shown to be valid for a Coulomb like potential. However, the statement of this theorem is not, in general, valid for more singular potentials, such as V (z) ∼ z −2 .
Theorem IV: The ground state energy is a monotonically decreasing function of z Λ .
Proof: To prove this theorem, we will consider two cutoffs z 1 and z 2 , such that z 2 > z 1 .
Let us denote by ψ 0 (z, z 1 ) the ground state wave function for the cutoff z 1 . As before, ψ 0 (z, z 1 ) is a real analytic function of z for all 0 < z < z 1 . Let us denote the corresponding case, for all z Λ > 0 all eigenvalues M 2 = E k z Λ > 0. Hence, there are no tachyons and the model is stable. The second possibility is that there is a critical value z c , such that when z Λ = z c we have E 0 z Λ = 0. Then, according to Theorem IV, whenever z Λ > z c there must be at least one solution with M 2 = E < 0.