Higher-Derivative Supergravity and Moduli Stabilization



Higher-Derivative Supergravity and Moduli Stabilization

David Ciupke, Jan Louis, and Alexander Westphal

Deutsches Elektronen-Synchrotron DESY, Theory Group, D-22603 Hamburg, Germany

Fachbereich Physik der Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

Zentrum für Mathematische Physik, Universität Hamburg,

Bundesstrasse 55, D-20146 Hamburg, Germany,,


We review the ghost-free four-derivative terms for chiral superfields in supersymmetry and supergravity. These terms induce cubic polynomial equations of motion for the chiral auxiliary fields and correct the scalar potential. We discuss the different solutions and argue that only one of them is consistent with the principles of effective field theory. Special attention is paid to the corrections along flat directions which can be stabilized or destabilized by the higher-derivative terms. We then compute these higher-derivative terms explicitly for the type IIB string compactified on a Calabi-Yau orientifold with fluxes via Kaluza-Klein reducing the corrections in ten dimensions for the respective Kähler moduli sector. We prove that together with flux and the known -corrections the higher-derivative term stabilizes all Calabi-Yau manifolds with positive Euler number, provided the sign of the new correction is negative.

May, 2015

1 Introduction

In many applications supersymmetric field theories or supergravities are considered as an effective description of a more fundamental theory, such as string theory. Most properties of this low energy effective theory are captured by the leading two-derivative Lagrangian . It can, however, happen that specific couplings vanish in and then higher order corrections do become important. A particular class of corrections are higher-derivative terms which in supersymmetric theories can simultaneously induce corrections of the scalar potential. It is the purpose of this paper to analyse supersymmetric higher-derivative operators with this property – both conceptually and as a new tool to stabilize moduli in string theory. Such terms were also studied in [1, 2, 3, 4, 5], while [6, 7, 8, 9, 10] started looking at their implications for cosmology.

More precisely, we focus on supersymmetry and supergravity in four space-time dimensions and within such theories on ghost-free higher-derivative operators. In non-supersymmetric theories it is well-known that the unique ghost-free four-derivative operator for a scalar field is given by .1 Several distinct superspace-operators exist which induce such terms. However, there is a unique ghost-free operator given by [2]


where denote the superspace derivatives, denotes the integration over the Grassmann variables and is a chiral superfield. We will see that the equation of motion for the auxiliary field is cubic instead of linear after including . This in turn implies up to three inequivalent solutions for and, hence, three inequivalent on-shell theories. The presence of this multiplet of theories is somewhat puzzling as one seems to loose predictability. However, studying the explicit solutions we find that only one out of the three theories is consistent with the principles of effective field theory (EFT).

There is a notable example in which higher-derivative operators such as have been computed from radiative corrections in a manifest off-shell scheme, namely the effective one-loop superspace Lagrangian of the Wess-Zumino model [13, 14, 15]. These references focused purely on those higher-derivative operators that contribute to the scalar potential and in [15] an infinite tower of such higher-derivative operators, denoted as the effective auxiliary field potential (EAFP), was explicitly computed. To lowest order in superspace-derivatives this EAFP coincides with given in eq. (1.1). The full non-local EAFP turns out to imply a unique on-shell theory. When truncating this EAFP to a finite number of terms, the truncation naively produces multiple on-shell theories. Applying the truncation at higher order even increases the number of solutions. However, we will show that at any order of the truncated EAFP there is a unique Lagrangian which reproduces the dynamics of the non-local theory at that order and which is consistent with the principles of EFT. The remaining theories can be regarded as artefacts of the truncation of the infinite tower of higher-derivative operators similar to the emergence of ghosts in truncated theories [16].

Apart from addressing this conceptual issue we proceed to compute the on-shell Lagrangians for models with arbitrarily many chiral superfields both in global and local supersymmetry. In particular we focus on the induced correction to the scalar potential and analyze the situation where the two-derivative theory has a minimum with a flat direction which can (or cannot) be lifted by the presence of .

In the second part of this paper we will purely focus on the effective action obtained from type IIB flux compactifications on Calabi-Yau orientifolds. The background fluxes are able to stabilize the complex structure moduli and the dilaton [17, 18]. In contrast, all Kähler moduli are described at leading order by a no-scale supergravity and thus are flat directions of the potential. Perturbative corrections for the Kähler moduli are induced from - and -corrections in the ten-dimensional action. An important example is the leading order -correction to the Kähler potential which is computed by reducing higher-curvature terms in ten dimensions [19]. This correction breaks the no-scale property, but by itself does not lead to a stabilization. When non-perturbative effects are taken into account scenarios with supersymmetric [20] or non-supersymmetric minima can be found [21].2 There is an intrinsic merit to demonstrate the existence of various classes of meta-stable de Sitter (dS) vacua as explicitly as possible in well-controlled examples of string compactifications. Thus, we find it worthwhile to explore further possibilities of moduli stabilization using only fully perturbative and explicitly computable contributions.

It is thus of interest to pursue the question to what extent additional -corrections of the ten-dimensional theory can lead to a stabilization of moduli without taking into account non-perturbative effects. Indeed, there are several such contributions which have not been discussed in detail, owing to the fact that the explicit structure of many of these terms is still unknown. It turns out that these terms do not correct the Kähler potential of the four-dimensional action, but instead require the presence of higher-derivative operators such as as off-shell completions. At this point the results of the first part of the paper can be used since precisely links four-derivative terms to corrections of the potential. By computing the four-derivative terms from the explicitly known -terms in ten dimensions [25, 26], the correction to the potential can be indirectly inferred. We find


where the denote the two-cycle volumes, the overall volume and the are topological numbers defined as


They encode information of the second Chern class and form a basis of .3

We then proceed to study the minima of taken together with the potential obtained from the -corrected Kähler potential. We show the existence of a model-independent non-supersymmetric minimum of this potential where all four-cycle volumes are fixed to values for any Calabi-Yau threefold with .4 This result suggests the existence of many new non-supersymmetric vacua within the landscape, where stabilization occurs purely from the leading order -corrections, but a more detailed discussion of all possible -corrections will be necessary to support this. Furthermore, the minimum only exists if the overall sign of is negative. This sign is universal and does not depend on the choice of the Calabi-Yau. Unfortunately, determining this sign requires the knowledge of the particular linear combination of all additional 4D higher-derivative operators contributing to the 4D four-derivative kinetic terms. This is beyond the scope of this paper and we leave it for future work.

This paper is organized as follows. In section 2 we study in effective theories with global supersymmetry. The conceptual discussion of the on-shell theories is performed for theories with a single chiral superfield in section 2.2 and in appendix A, where we also display the exact solutions for the chiral auxiliary field and prove the absence of ghosts. In section 2.3 we illustrate the interpretation of the higher-derivative operators and the respective on-shell theories with the one-loop Wess-Zumino model. In section 2.4 we then display the physical on-shell Lagrangian for arbitrarily many chiral superfields and make some statements regarding the structure of the resulting minima, providing an explicit example for the lifting of flat directions in section 2.5. In section 3 we show the respective Lagrangians for the case of supergravity and again discuss the structure of the minima with an explicit example in section 3.2. Finally in section 4 we turn to the discussion of flux compactifications of Type IIB on Calabi-Yau orientifold, where the details of the reduction of the curvature-terms in ten dimensions can be found in appendix B and appendix C. At the end we provide some conclusions in section 5.

2 Higher-Derivative Terms in Supersymmetry

2.1 Preliminaries

In this section we consider globally supersymmetric theories with chiral superfields ,   whose couplings are encoded in a Kähler potential , a superpotential and the higher-derivative operator . In the following we adopt the conventions and notation of [27]. Thus, the total superspace Lagrangian is of the form5


In the spirit of [3] we allow for an arbitrary hermitian four-tensor superfield which we assume to depend only and but not on any derivative.6 We will often refer to this mass dimension quantity, respectively its scalar component as coupling tensor. From the structure of one infers the symmetry properties


In order to obtain the component expression of we use the well known -expansion of the chiral superfields


where are scalars, chiral fermions and auxiliary components. From the form of the superspace derivatives


one finds that the bosonic part of only has a contribution at order which is given by


Performing the integration in eq. (2.1) one obtains the Lagrangian


where and denotes the holomorphic derivative of the superpotential. We indeed see that no derivative terms for appear and, thus, their equations of motion stay algebraic such that the remain non-propagating auxiliary fields. However, contains quartic terms in the which lead to cubic contributions to the bosonic part of the respective equations of motion


Determining all solutions to this equation in all generality is a delicate task and therefore we first turn to a theory with a single chiral multiplet where we can solve the cubic equation (2.7) exactly.

2.2 Theory with one Chiral Multiplet

For one chiral multiplet eqs. (2.7) reduce to


where we defined for brevity. In appendix A.1 we solve eq. (2.8) exactly and show that depending on and the specific region in the phase space of one or three solutions for exist. Expanding the solutions for small and inserting into eq. (2.6) keeping only the leading terms one obtains, in the case where all three solutions exist, the following three Lagrangians


where for convenience we defined and is the scalar potential of .7 In the following we will sometimes refer to the individual Lagrangians in eq. (2.9) as branches. We observe that is analytic in and reproduces at leading order. At linear order in it induces a correction to the kinetic energy, which is proportional to , as well as to the potential, proportional to . on the other hand have a pole-like term in and at order only have a contribution to the potential, which differs from by a factor .

In summary the theory defined by (2.1) can lead to three different and independent on-shell Lagrangians. However, a multiplet of theories is dissatisfying, since it predicts several inequivalent evolutions of fields for a given set of initial data. Furthermore, suppose we include additional off-shell higher-derivative operators with more than four superspace-derivatives then the equations of motion for the chiral auxiliaries admit more than three solutions, rendering the problem even more severe. Let us now argue how to resolve this issue in the context of an effective field theory.

When performing the limit in the off-shell Lagrangian given in eq. (2.1) we recover the ordinary, two-derivative theory . For consistency this should also hold in the on-shell theories given in eq. (2.9). For example suppose that the higher-derivative operator arises by integrating out massive states associated with a mass scale from a UV theory. Then to lowest order in fields one has and hence the operator should decouple as becomes large compared to the masses of the light states as dictated by the decoupling principle, see for instance [28]. We see that given in (2.9) is analytic in , while contain a non-analytic part and thus violate the decoupling limit. Based on this observation we propose to regard only as the physical on-shell Lagrangian since it is the unique Lagrangian compatible with the principles of effective field theory. We will substantiate this proposition with the example of the effective one-loop Wess-Zumino model in the next section. Notably we will show that the non-analytic theories not only fail to obey the decoupling limit, but furthermore are incapable of reproducing the on-shell Lagrangian of the full, non-local theory. To some extent this is already visible in eq. (2.9). More precisely the non-analytic branches fail to reproduce the terms in . In fact they neither include the kinetic terms nor the scalar potential of . On the other hand the contributions in exactly coincide with the terms in . In summary, this observation and the results of the next section suggest that the non-analytic solutions should be regarded as mere artefacts of the truncation of an infinite sum of higher-derivatives. Note that the above observation is reminiscent of the discussion of theories with higher-derivative terms in the equations of motion where ghost-like degrees of freedom emerge. Similarly the ghosts arise from truncating an infinite series of higher-derivative terms to a finite sum and violate EFT-reasoning in as much as the inclusion of higher order operators should merely induce a small correction to the dynamics of some IR-Lagrangian. A ghost-free theory can then be obtained by demanding analyticity of the solutions to the equations of motion in EFT-control parameters [29, 16], identical to our reasoning above.

In the rest of this paper we will therefore only discuss the analytic theory. Furthermore, recall that besides the operator in eq. (2.1) superspace higher-derivative terms with more than four superspace-derivatives exist and they contribute higher polynomial powers of the auxiliary field to the Lagrangian (next section we display the one-loop Wess-Zumino model as an explicit example where infinitely many superspace-derivative operators are present). These operators are further mass-suppressed and hence modify the equations of motion for the auxiliary fields at order .8 This implies that without including such higher-derivative terms into the superspace Lagrangian, we can trust the resulting on-shell Lagrangian only up to linear order in .9 Fortunately this greatly simplifies the structure of the on-shell Lagrangian and makes a proper discussion of the multi-field case feasible.

To conclude this section let us describe why the theory is free of ghosts. The absence of ghosts is not immediately clear, but can be understood with the exact solution for the auxiliary field at hand. The sign of the ordinary kinetic term is affected by the presence of the higher-derivative operator through eq. (2.8). In appendix A.2 the absence of ghosts is explicitly demonstrated for the theory obtained by solving eq. (2.8) exactly and reinserting the result into eq. (2.6). Nevertheless, one might still worry about the sign of the ordinary kinetic term in the truncated theory after inspection of eq. (2.9). More precisely one finds that the theory becomes ghost-like once . However, in that regime we cannot trust our truncation at linear order in any longer as we illustrate in appendix A. In other words, studying the exact solutions of eq. (2.8) shows that if , the analytic solution ceases to exist and one enters a regime, in which only non-perturbative solutions can be found. To summarize, the analytic theory breaks down before it would become ghostlike.

2.3 One-loop Wess-Zumino Model

After the general discussion of the previous section let us now turn to an explicit example, where the truncation of the infinite sum of higher-derivatives and the structure of the equations of motion for the auxiliary field can be explicitly studied. This example is given by the one-loop Wess-Zumino model in superspace, for which the full, non-local effective auxiliary field potential (EAFP) was recently computed in [15] following up on earlier works [13, 14]. More precisely the model consists of a single chiral superfield with Kähler potential and superpotential of the form


According to [15] the only contributions to the effective superspace potential at one-loop come from corrections to the Kähler potential as well as an EAFP, which we denote as . More precisely it consists of an infinite tower of higher-derivatives of the form


where and is a known real-valued analytic function with non-vanishing coefficients in the respective series expansion at all orders [15]. The lowest order contribution arises from the constant term in the series expansion of and comparing with (2.1) we have


Expanding as a geometric series, we identify that to lowest order we have .

Let us now proceed by performing the superspace integration in eq. (2.11). From eq. (2.5) we infer that the bosonic part of the superfield multiplying has only a contribution and hence the remaining superfields have to be evaluated at their scalar component. This yields


For simplicity let us set from now on. displays an infinite sum in the auxiliary field and . Additional powers of the auxiliary field are in a one-to-one correspondence with additional powers of superspace-derivatives. We can identify


as the parameter controlling the infinite series of higher-derivatives and powers of the auxiliary field, respectively. We immediately observe that eq. (2.13) comprises an analytic function in . Using the full (and explicitly known) function it can be numerically shown that the solution to the equations of motion for derived from the standard Lagrangian plus is unique and analytic in .

The non-local theory with in eq. (2.13) can be regarded as a UV-theory for a local theory after truncating the infinite sum of higher-derivatives to a finite sum. For the purpose of obtaining a local theory also the control parameter has to be truncated. However, we omit this here, as it does not provide additional insight into the structure of the series in higher-derivatives.

It is interesting to discuss the equations of motion for the auxiliary field once the theory is truncated at a given order in . In the following let denote the truncation of the series expansion of at order . If we truncate at , the discussion reduces to the familiar cubic in eq. (2.8), which admits only one analytic solution. For arbitrary the contribution of eq. (2.13) to the scalar potential reads


Taking into account the remaining, ordinary terms in the Lagrangian, i.e.  in eq. (2.1), the equation of motion for reads


where we only took into account terms that contribute to the scalar potential. induces monomials in up to degree and, hence, eq. (2.16) admits up to independent solutions. In other words the number of solutions is increasing with the order of the truncation. To solve eq. (2.16) we first redefine the auxiliary field via


Inserted into eq. (2.16) one observes that has to be real and, hence, eq. (2.16) reduces to


We make an ansatz of the form


such that eq. (2.18) at lowest order in reads


Since is a polynomial of degree with non-vanishing coefficients we see that only the branch given by is analytic. All other solutions, which are defined at lowest order by the remaining solutions of eq. (2.20) and necessarily fulfill , are non-analytic in for any .

In effective field theory one generally expects to be able to compute observables with higher precision by including more and more operators. Indeed since the unique solution of the non-local theory was analytic, the analytic solution of the truncated theory is able to reproduce the Lagrangian of the non-local theory at order and, thus, mimics the non-local theory with better precision for larger . However, regardless of the order of the truncation the non-analytic theories fail to reproduce the non-local theory to that specific order. One can explicitly check this for the first components in the expansion in eq. (2.19). At lowest order this was also already visible in eq. (2.9).

It is worth noting that the existence of a unique analytic solution for in the truncated theory does not depend on the details of the , but we expect it to hold in general as long as the coefficient of the term in the Lagrangian is non-vanishing. Indeed the EAFP is correcting the Lagrangian by at least cubic powers of and [13] so that one would always expect the analytic solution to be unique.

After the above conceptual discussion we can now proceed to study theories with more than one chiral multiplet.

2.4 Multi-Field Case and Analysis of Scalar Potential

Given the results of the previous sections we constrain the discussion of the multi-field case to the analytic solution of eq. (2.7). Solving eq. (2.7) using perturbation theory yields at linear order in


Insertion of the auxiliary field into the Lagrangian in eq. (2.6) yields


The resulting scalar potential at linear order in reads


Before we analyse this potential, let us make a comment regarding the ordinary kinetic term in the Lagrangian in eq. (2.22). The metric multiplying the kinetic term is corrected by


In general it is not possible to absorb the correction in eq. (2.24) by performing a change of coordinates in field-space and, hence, the metric multiplying the kinetic term in eq. (2.22) is in general not a Kähler metric.10 For the following special form of the coupling tensor


with this was demonstrated explicitly in [1].

Since the supersymmetry transformations of the chiral multiplets do not change, the order parameter for supersymmetry breaking continues to be . Therefore the supersymmetric minima of are found at


From eq. (2.7) we see that the supersymmetric locus in field space which solves (2.26) is determined by and, thus, is not corrected by the presence of the higher-derivative terms under the condition that is non-singular.11 Indeed it was shown that for arbitrary higher-derivative theories the structure of the supersymmetric vacua is unchanged [1]. In particular this implies that any flat direction of is not lifted.

If supersymmetry is broken by some the higher-derivative correction can become important. Still is a perturbation of and therefore the minimum of will at best be shifted to a nearby field value . However, if the non-supersymmetric minimum of has a flat direction the contribution from becomes the leading term in this direction and may lift its flatness. A possible exception to this occurs when the flatness is due to a symmetry, such as a perturbatively unbroken shift-symmetry. Further exceptions are models in which supersymmetry breaking occurs due to a spontaneously broken R-symmetry [30]. In this case there always exists a flat direction, the R-axion, associated with the Goldstone boson of the broken R-symmetry. Here the existence of higher-derivative corrections does not lift the flatness.

If the flatness is lifted, then depending on the structure and sign of the flat direction can be stabilized or destabilized. It is difficult to make a general statement, and in the end a case-by-case analysis is necessary. Nevertheless, before we proceed, let us offer some general observations.

A (real) flat direction is characterized by the fact the all -derivatives of vanish in the background, or in other words


Let us assume that has a flat direction and thus satisfies (2.27). A special (and simple) case of this situation is that does not depend on at all, i.e. . In this case the flat direction is lifted for generic but preserved if is also independent of . A slight generalization occurs when and only the matrix element of in the direction of the supersymmetry breaking -term, say , are independent of . In this case the flat direction is preserved if also is independent of . As a final example let us discuss a specific form of the coupling tensor given in eq. (2.25). In this case we have and thus any flat direction of remains flat with respect to , given that the scalar function does not depend upon it.

2.5 Example: O’Raifeartaigh Model

For concreteness let us discuss a specific example of a model with flat directions within non-supersymmetric vacua. The simplest case is given by the O’Raifeartaigh model. This is defined via a Kähler and superpotential, which read


Here are real parameters such that . The resulting potential is minimized at leaving unfixed. Since , supersymmetry is broken in the vacuum. Eq. (2.28) has a -symmetry in and and furthermore an R-symmetry, if we assign R-charges as follows


For the continuum of vacua labeled by there exists one vacuum, namely , in which the R-symmetry is not spontaneously broken. Thus, the O’Raifeartaigh model is an exception to the generic expectation that supersymmetry breaking occurs due to R-symmetry breaking in models, which reduce to Wess-Zumino models in the low energy regime and respect the principles of EFT [30].

Let us proceed by switching on the higher-derivative operator. We consider vacua in which as in the ordinary theory. The respective potential at the point is extremized, if the following holds


We see that the flatness of is lifted, if certain components of the tensor require a specific value for extremization.

Inspecting eq. (2.1) we find that the higher-derivative Lagrangian is R-symmetric, if


The most general coupling tensor at quadratic order in fields respecting the - and R-symmetry is given by


For simplicity we suppressed the tensor indices of and here. From eq. (2.30) we see that is fixed in the minimum to the value , in which the R-symmetry is preserved, unless the following couplings vanish


In a generic effective field theory there is no reason why these couplings could be zero and so one concludes that indeed is fixed. Note furthermore that if the R-symmetry would have been broken in the minimum, then a flat direction associated with the respective Goldstone boson would have persisted. Finally, note that the flatness of can also be lifted by including higher-dimensional operators into the Kähler- or superpotential.

3 Higher-Derivative Terms in Supergravity

3.1 Preliminaries

Let us now couple the theory specified in (2.1) to supergravity. We will only reproduce the essential steps here and refer the reader for a detailed derivation to the original paper [3]. Without any higher-derivative operator the Lagrangian is given by [27]


where denotes the chiral density, the curvature superfield and with being the covariant spinorial derivative. To obtain the Einstein-frame Lagrangian for the scalar fields , it is necessary to perform a Weyl transformation of the vielbein and successively integrate out all the auxiliary fields. This results in the familiar scalar potential


where is the Kähler covariant derivative of the superpotential.

To couple the higher-derivative operator of eq. (2.1) to supergravity one can either add the term [3]


to (3.1) or modify the Kähler potential as12


Due to (2.5) the bosonic Lagrangians obtained by the two methods coincide up to a Kähler factor, which can be absorbed in a redefinition of . Here we assume that only depends on the chiral and anti-chiral superfields and but not on the gravitational multiplet.

In the Lagrangian one performs the same Weyl-transformation as before and integrates out the auxiliary fields in the gravitational multiplet. This procedure is not affected by the presence of . One is then left with the Lagrangian [3]


The equations of motion for now read


After the discussion in the previous section we only focus on the analytic solution of (3.6).13 Here it is sufficient to know the auxiliary fields up to linear order in the coupling tensor. They read


Inserting the above auxiliary field into the Lagrangian in eq. (3.5) yields


The scalar potential is corrected as follows


where is given in (3.2) while


Analogous to eq. (2.22) the metric multiplying the ordinary kinetic term receives a correction. From eq. (3.8) we read off its form


As in the global case this correction in general renders the metric non-Kähler.

3.2 Fate of Flat Directions and Simple No-Scale Examples

Let us begin the analysis with the supersymmetric minima of the potential given in (3.2),(3.9) and (3.10). denotes the order parameter for supersymmetry breaking. Analogous to the discussion with global supersymmetry eq. (3.6) implies that unbroken supersymmetry imposes the exact same condition as in a standard two-derivative supergravity, that is


Thus, the location of the supersymmetric minima in field space are determined by and they are unaffected by the presence of . In particular, any flat direction of is preserved by . In addition, corresponds to a Minkowski vacuum while corresponds to an AdS vacuum.

Let us now turn to minima with spontaneously broken supersymmetry. As in the global case is considered to be a perturbation of and the minimum of is shifted to a nearby field value . Therefore qualitatively nothing changes except for the flat directions. Contrary to the case of global supersymmetry in the local case non-trivial models with vanishing potential exist. These are the no-scale models. The no-scale property is generally expected to be lost when higher-derivative corrections are taken into account, thus making it possible to lift flat directions. In the rest of this section we present a simple example to illustrate the fate of flat directions and make a first step towards the potential relevance to moduli stabilization.

More precisely we consider a model specified by a constant superpotential and the Kähler potential


where . This is of the no-scale type in that it satisfies


In this case given in (3.2) is positive (negative) for and vanishes identically for . Adding given in (3.10) and redefining one obtains


For both real and imaginary parts of are flat directions of . We see that generically both flat directions are lifted unless the combination is constant in and/or . For example a continuous shift symmetry i const. which often holds perturbatively in string theory would protect the flat direction along in that could not depend on . In order to say something about the stability, however, one has to make some assumptions about the functional dependence of .

Let us now consider a very simple situation, in which the inclusion of stabilizes a certain direction. For instance if and const.,14 the two terms in eq. (3.15) can balance for with a non-supersymmetric AdS minimum at


Furthermore we have to check whether the field-value in eq. (3.16) is within the regime, where the perturbative solution for the auxiliary field converges. An estimate for the boundary between the perturbative and non-perturbative regime can be obtained from the results of appendix A. Indeed, from eq. (A.11) one infers that the boundary lies at


We see that has to be sufficiently large for some given