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Anti-brane uplift instability from goldstino condensation

We investigate the possible appearance of composite states of the goldstino in models with four-dimensional non-linear supersymmetry and we provide a description of their dynamics in terms of a K\"ahler potential and a superpotential. Our analysis shows that the critical point corresponding to the Volkov-Akulov model is unstable. Similarly, we find that the uplifted stable de Sitter critical point of the KKLT model is shifted and acquires a tachyonic instability. Our findings indicate the existence of a potentially dangerous instability shared by all anti-brane uplifts.


Introduction
One of the central ingredients in typical string theory de Sitter constructions is the use of anti-D3-branes to uplift an AdS vacuum to a de Sitter critical configuration [1][2][3][4][5][6]. Even though the end result of such uplift is often challenged [7], and the existence of tachyons or loss of criticality is implied [8][9][10][11], there is no conclusive indication that the four-dimensional effective field theory suffers from the alleged instabilities. (Recent selected possible issues from a tendimensional perspective are further reported in the articles [12][13][14][15][16][17][18][19]). If one assumes that there is a consistent low energy effective field theory that can incorporate the anti-D3-brane uplift, then one expects to be able to embed it within 4D N = 1 Supergravity with the use of a nilpotent chiral superfield that breaks supersymmetry and provides the uplift [20][21][22][23][24][25]. Indeed, the embedding of the KKLT-type uplift within Supergravity coupled to a nilpotent superfield was discussed in [26,27], and the underlying non-linear supersymmetry of the anti-brane uplift was made manifest. More generally, the appearance of a goldstino sector on anti-Dp-brane worldvolumes was also investigated in [28][29][30][31].
If the aforementioned simple de Sitter constructions are indeed unstable, one of the cleanest ways to see this would be to uncover this instability in the low energy 4D N = 1 supergravity description. To support this approach, let us observe that there do exist hints that the Volkov-Akulov (VA) model coupled to pure 4D N = 1 Supergravity [32,33] is inherently unstable. This is seen by considering a real scalar representing a gravitino condensate ψ m ψ m ∼ σ(x), and studying the effective theory à la Nambu-Jona-Lasinio [34,35] (also similarly to composite Higgs models [36]) for the four-gravitini interaction. Such an analysis has been performed in the early supergravity bibliography in [37,38] to yield the effective scalar potential, but the actual existence of a condensate was later questioned [39,40]. More recently, the condensation was revisited and put on firmer footing in [41][42][43]. Furthermore, both the Fierz ambiguity and the wavefunction renormalization of the condensate were addressed [44]. A comprehensive and analytic review of all this work can be found in [45]. The crucial result in these articles pertains to the behaviour of the effective scalar potential for the condensate σ(x), which turns out to be tachyonic around the original central vacuum. This, therefore, indicates that an uplift from an AdS vacuum to a de Sitter one with a pure non-linear realization may be inherently unstable.
The drawback of these articles, however, is that a manifestly supersymmetric analysis is missing and a precise and controlled form of the effective theory for the condensates is elusive. This also means that, if one wanted to extend the analysis to a matter coupled Supergravity, for example to the Kähler modulus of KKLT, the full procedure would have to be performed from scratch.
In this work we make a first step towards filling this gap by providing a manifestly supersymmetric description of the composite states. One could suspect that the effect that manifests itself in a supergravity framework as gravitino condensates may already exist in the rigid limit in the form of goldstino condensates. Indeed, such behaviour of the Volkov-Akulov fermion can be justified by the fact that the theory contains four-Fermi interactions, just like the fermions in the Nambu-Jona-Lasinio model. In a typical Nambu-Jona-Lasinio setup the way in which the effective theory for the bound states is uncovered is by recasting the four-Fermi interactions ΨΨ 2 as −σ 2 + σΨΨ, with σ taken to be auxiliary at some UV scale, and then following the flow of the theory to the IR. This generates a kinetic term for the scalar σ and also gives rise to new contributions to its effective potential, which leads to the formation of a new vacuum where the condensation takes place. However, for a single goldstino, a large-N expansion that helps to control the diagrams in the Nambu-Jona-Lasinio model is not available; as a consequence, a perturbative loop-diagram analysis is not tractable and a direct non-perturbative analysis is required. Indeed, the Nambu-Jona-Lasinio model can be discussed for a single fermion species if a non-perturbative renormalization group flow is utilized [46].
Our procedure here is thus the following. We first recast the full Volkov-Akulov model in terms of two unconstrained superfields, X and T , where the latter is a Lagrange multiplier that is integrated out to impose the nilpotency condition X 2 = 0 on the former [47][48][49]. Once the nilpotency is imposed, we recover the typical Volkov-Akulov model. To uncover the low energy description of the theory, and possibly the existence of light composite states, we have to track the flow of the theory towards the IR. We do this by following the flow determined by the exact renormalization group (ERG) equations within a supersymmetric rendition of the local potential approximation (which we will denote as SLPA) [50][51][52][53][54][55][56]. In particular, we perform this analysis preserving supersymmetry off-shell and we keep track of the full flow of the Kähler potential.
Our final result verifies the emergence of composite states, which now fall into standard chiral multiplets with linearly realized supersymmetry, with an effective low energy description where the Volkov-Akulov Kähler potential K VA = XX is replaced by K composite states = Z X XX + Z T T T + higher order terms , where the Z s indicate the wavefunction renormalizations, and where the superpotential remains the same as in the original Volkov-Akulov model, A study of the potential of the resulting supersymmetric theory shows that tachyons are generated near the origin of the (X,T ) field space, signaling an inherent non-perturbative instability of the pure Volkov-Akulov model. The compositeness of T is manifested by the vanishing Z T at the UV point of the flow 1 .
A further step would be to perform a similar procedure in Supergravity using again an ERG flow [59,60]. An important obstacle to performing this analysis in Supergravity is the requirement of a supersymmetric regulator. However, for small fields, one can trust the supersymmetric analysis: the supergravity effects should only enter as we probe larger distances in field space, where the 1/M P effects will become important. Therefore, the same low energy effective description for the Volkov-Akulov composite states can be justified to hold also in Supergravity, but only near the field space origin. We use this approximation to show that the Volkov-Akulov model, now coupled to 4D N = 1 Supergravity as in [23], again suffers from tachyonic instabilities; this verifies the earlier results regarding tachyons due to gravitino condensation [38,44].
The same holds also for the KKLT supergravity embedding, when we introduce a single Kähler modulus [26]. Our results actually indicate that, if a 4D N = 1 supergravity theory flows in the IR to the model of [23], then it will suffer from the same instability (as it happens for the KKLT 4D N = 1 embedding of [26]). The only way to avoid the instability associated with the formation of these composite states would be to always have some additional light states (possibly of non-perturbative origin) surviving in the IR and either disrupt the procedure that we described or alter the flow.
2 Composite supersymmetry from the Volkov-Akulov model

The setup
The four-dimensional Volkov-Akulov model [32] can be described in terms of a constrained superfield X that satisfies [47,48] with the Lagrangian, defined in terms of a Kähler potential K and a superpotential W , This procedure is further described in [49] and the component form of the Lagrangian is (using the mostly-minus convention for the spacetime metric, η mn = diag(+1, −1, −1, −1)). The presence of the non-linear self-interaction terms opens the possibility of composite states of two goldstini, e.g. G 2 /f . We wish to investigate the possible description of such states in the low energy theory 2 .
First, we bring the theory into a form where supersymmetry is linearly realized, and use it as the form of the theory at the UV point. To this end we introduce a Lagrange multiplier multiplet T (associated with no term in the Kähler potential) by changing the superpotential to By varying T we recover the superspace condition (2.1). Note that the Volkov-Akulov model naturally comes with a UV scale given by the supersymmetry breaking scale, which is controlled by f ; it is therefore natural, though not mandatory, to match the two descriptions at that energy scale.
At this point, let us prove that (2.4) is the most general form of the superpotential that one can write in the UV point and that K = |X| 2 is the most general Kähler potential.
We firstly observe that, by definition, the Kähler potential has no dependence on T in the UV point. We then assume that the superpotential takes the form W = f X + 1 2 T X 2 + P (X), where P (X) = n≥0 P n X n = P 0 + P 1 X + P 2 X 2 + P 3 X 3 + . . . is some arbitrary analytic function (as it can always taken to be, since we are dealing with a superpotential), and we further note that we can not have terms like 1/X p with p > 0, because they will become ill-defined once the nilpotency condition is imposed. Now, P 0 clearly drops out due to the d 2 θ and P 1 can be absorbed into f . So, we remain with f X + 1 2 T X 2 + X 2P (X), whereP (X) = P 2 + P 3 X + P 4 X 2 + . . . , which is still granted to be an analytic function of X. If we simply shift T to T − 2P (X), we are left with our original superpotential (2.4). This shift is consistent precisely because the functionP (X) is analytic. Moreover, there are no T -dependent terms in the Kähler potential, so this shift does not generate new terms.
Analogously, let us consider the Kähler potential K = |X| 2 + m,n≥0 M nm X n X m assuming by fiat that it can be expanded in a power series (as part of our general assumptions). We can directly see that M 00 , M 01 and M 10 drop out due to the d 4 θ, whereas M 11 is absorbed by redefining the |X| 2 term. Then, we are left with whereM (X, X) = m,n≥0M nm X n X m . Now, once simply shifting T to T + 1 2 D 2M (X, X), we are left with our original Kähler potential.
We are thus working with the most general Kähler potential and superpotential at the UV point.
The second step of our procedure is to lower the energy scale at which we probe the theory via a renormalization group (RG) flow and this will make the superfield T acquire a kinetic term.
The interpretation of this is that the new standard chiral superfields X and T now describe composite states of the original fermion goldstino. This means that the partition function of the system has the form We will study the RG flow by using the exact renormalization group method discussed in [50,51,56]. To follow this method it is typical to introduce all possible terms in the Kähler potential and the superpotential that are consistent with the symmetries that are expected to remain preserved, such that the flow can be properly described. Some terms may stay trivially zero throughout the flow depending on the initial conditions and the subsequent flow. Since such a procedure actually requires to introduce infinite terms, in order to make it tractable the local potential approximation (LPA) has been devised (see e.g. [52][53][54][55] Here, we will then use a type of local potential approximation that allows us to preserve supersymmetry manifestly, which we will simply call, as already mentioned in the introduction, the supersymmetric local potential approximation (SLPA). It is easier to describe this approximation directly in the superspace language 3 . First of all, supersymmetric theories with chiral superfields X i are in any case described by two types of superspace integrals, which are d 4 θ and d 2 θ. If we think of the functions that we can insert in those integrals as having a (superspace) derivative expansion, then we can have where the first term corresponds to the Kähler potential and the second one to the superpotential.
Our SLPA can be now simply defined by stating that we will not keep track of superspace higher derivative terms. This means that any term in (2.6) that is not a a part of K or W will always be ignored. Similarly, since we will be working in component form, we will not keep track of any terms that do not correspond to a Kähler potential or a superpotential. We will ignore these terms when they are generated during the flow and we will not take into account their backreaction into the ERG equations.
Even though these types of approximations are in standard use in the ERG literature, it is important to understand and be conscious of their limitations regarding the physical conclusions that one can draw. We will discuss these issues carefully in the next section. For now, we proceed within this approximation and derive the SLPA RG flow of our model. We will, however, note immediately that in our chosen approach, even though we will not be able to see the anomalous dimensions of the fields, we will have a certain handle on the wavefunction renormalizations because of the presence of other fields besides the physical scalars.
Even within the LPA, one still makes an expansion in the fields to derive the RG flow, which means that we still need to make an expansion in terms of superfields in the Kähler potential. In our case, we will make an educated guess and keep only the terms that provide a self-consistent flow. Indeed, the gratifying result of applying the SLPA is that we get a solution to the ERG in a closed form that includes only a handful of new terms in the Kähler potential. As we will see, it will suffice to work with the Kähler potential and the superpotential [q] = −2. This theory is defined with a UV cut-off Λ. We want to study its properties as we integrate out the high energy modes down to a lower scale µ, with This is often called the renormalization scale and it is related to the energy scale at which we probe the theory. Then, the renormalization time is defined as 10) and the flow to the IR is described by µ ↓ and t ↑. To match with the Volkov-Akulov action at the UV point, that is when µ = Λ, we want to find the RG flow of the couplings with the boundary conditions Indeed, we notice that β(µ = Λ) = 0 and g(µ = Λ) = 0 are required in order for the superfield T to act as a Lagrange multiplier at the UV scale and q(µ = Λ) = 0 because, as we have showed above, K = |X| 2 is the most general expression of the UV Kähler potential that we can have.
Following [50,51,56], we re-organize the action into propagator and interaction parts. We do this in a supersymmetric manner by appropriately splitting the Kähler potential. In particular, we have where c is a scale-dependent regularization function, which we will discuss momentarily, with [c] = 0, and Note that we leave the background field dependence in the interaction part of the Kähler potential. Similarly, the superpotential naturally only contributes to the interactions: (2.14) Moreover, since no background-independent mass term exists neither for X nor for T in the UV, and it will not be generated during the flow 4 , we do not include a mass for these fields in the propagator piece (2.12).
Returning to the propagator regularization function c, working in Euclidean momentum space, we can write as long as some basic asymptotic properties are satisfied [56], where now we have |p| ≤ 1.
For our calculations we will not need to work with an explicit form for c, but for the benefit of the reader we can give as a simple example the expression c(p, µ) = 1 −p 2 Θ 1 −p 2 , which is discussed in [56]. Such a regulator is manifestly supersymmetric, because the component field propagator terms are described collectively by the superspace integral d 4 θ (c(−∂ 2 )) −1 |X| 2 + |T | 2 and the superspace derivatives (the D s) commute with any combination of spacetime derivatives.
In addition, since we will only be interested in the vacuum (in)stability of the theory, we will not include any source terms in our analysis or any derivative interactions. As discussed in [51], In our procedure we will rely heavily on supersymmetry and, because we will work directly in component form, this means that, within the SLPA, we will only evaluate the flow of the coefficients of the auxiliary field potential that are related to the Kähler potential. In other words, we will only keep track of the terms of the form and from these terms we will deduce the full flow of the Kähler potential. A similar procedure is used in [64] for the evaluation of the one-loop Kähler potential. For the study of the flow we will be using the Euclidean conventions of [65], where the Lorentzian conventions correspond to η mn = diag(+1, −1, −1, −1), which also matches with the conventions of [50].
We now go to components and work with the momentum space fields. We definê We also define dimensionless couplings via γ = µ 2 g(µ) and ζ = µ 2 q(µ) , where the dots stand for many other interaction terms and many terms including fermions. Here L int. is only a formal compact expression and it really means that we should treat all terms in the expansion in the form

(2.24)
For completeness, let us mention that the full Euclidean partition function is just as in [50]. As already mentioned, the symbol L clearly refers to an action, but we keep the notation of [50], using the symbol L instead of S. Note also, once again, that all fields, momenta and couplings are dimensionless.
We can now use the ERG equation from [50] to obtaiṅ ∂h a ∂h a + + fermion propagator terms .
(2.26) 5 To get to Euclidean momentum space, we first Wick-rotate and then we go to momentum space.
We have independent sums in the ERG equation (2.26) because all our scalars have diagonal propagator terms. Notice that in a slight abuse of notation, we denote by partial derivatives what should really be understood as variational derivatives, with a momentum matching δ-function, i.e., say, The lack of a (2π) 4 factor on the δ-function above is due to the fact that it is already included explicitly in the expression (2.26). This choice of notation and normalization, which will be used throughout the rest of this section, corresponds to the one in [50].
At this stage we have to insert the interacting action (2.24) into the ERG equation and equate term by term to deduce the flow equations. We stress that one only needs to look at the terms related to the auxiliary field potential, namely To this end the third line in (2.26) does not play any role as one can check by considering the fermionic contribution to the ERG (see e.g. [51]) and the fermion couplings to the auxiliary fields, which are only linear in the auxiliary fields (see e.g. [66]). Therefore, we will focus on the first two lines of (2.26) and we will only use the fermions as a cross-check.
One can easily prove that the superpotential does not receive any corrections within the SLPA by checking that no terms linear in the auxiliary fields are generated. This can be seen faster by writing (2.26) in a form where the scalars and the auxiliary fields are recast to be complex by a simple chain rule.

The RG flow within the supersymmetric local potential approximation
Our aim is to find the flow equations for the couplings α, β, ζ and γ.
Let us first look at the equation governing the flow of ζ. This means that on the left hand side of (2.26) we want to focus on the term On the right hand side we have the term and a variety of seemingly relevant terms related to the auxiliary field propagators as, for instance, the term (2.31) As we will now see, only the term (2.30) actually contributes to this part of the flow, whereas (2.31) contributes to derivative interactions. Indeed, up to an overall coefficient, (2.31) gives which means that this term contributes only to derivative terms: in fact, from (2.16) we have In a similar way, we can see that the only relevant part of (2.30) is given by the c 1 part of the We then evaluate and This means that in the compact notation (2.23) the relevant part of (2.26) takes the form where the −2ζ is due to the fact that ζ originates from a dimensionful coupling.
Similarly, for the coupling γ we are bound to geṫ In this way we see that the tree level interactions from the superpotential contribute to the flow of the higher order terms in the Kähler potential.
We will now analyze the equations that govern the flow of β. The relevant term of left hand side of (2.26) isβ and on the right hand side we have and a variety of seemingly relevant terms related to the auxiliary field propagators, as the term As we will see right away, the term (2.42) and other similar terms from the second line of (2.26) do not enter this part of the flow and can be safely ignored. Indeed, focusing on (2.42) we find , which means that (2.42) does not contribute to the flow of β; it contributes, instead, to the higher order derivative terms.
Therefore, the flow of β is controlled by (2.41).
We have two terms in (2.41), but their contribution is the same: we will work out one of the two terms and double the result. We have Now, we postulate that the momentum integral over p 3 takes a value N , which is regulatordependent, and is given by The exact value of N can thus be evaluated only once we have a specific regularization scheme at hand. We conclude that and, once we sum over both scalars and referring to the compact notation of (2.23), we find that the relevant part of (2.26) takes the formβB 2 A similar analysis for F 2 1 is bound to give the flow equation for the coupling α, which iṡ This completes the analysis of (2.26) and the reader can check that no other terms are required for the self-consistent flow of the auxiliary field scalar potential. This means that the Kähler potential will not need higher order terms and that our flow is exact.
We conclude that the quantum effects make the Lagrange multiplier superfield T become propagating and a non-ghost kinetic term requires N c 1 > 0, which is in accordance with the typical properties (2.52). Indeed, it is typical to have a regularization scheme where [56] For example, these conditions are satisfied for the probe regulator c(p) = 1 −p 2 Θ 1 −p 2 , which gives c 1 = −1 and N = − 1 32π 2 . Note that the condition N c 1 > 0 also ensures that the multiplet X doesn't become of ghost type in the IR. One can also work with different regulators c(p), and, as we have already mentioned, find similar results. For instance, one can use an analytic function of the form e −p 2 −4p 4 , which would give again c 1 = −1 and N −10 −3 . An analytic regulator, however, will always have some contribution from the high UV modes because its support goes up to infinity.
We will now shift to canonically normalized dimensionless superfields by redefining them as follows: µ being the scale compared to which we measure energies and lengths. In addition, for the action and the superspace integrals we will have where the new x and θ are dimensionless. This means that, when the redefined fields take a VEV, e.g. T = 0.1, the original field had a VEV of the form 0.1 × µ/ √ β. We thus obtain a Lagrangian where all fields are dimensionless and all couplings are dressed with µ, i.e. g will always appear in the combination µ 2 g (and analogously for q). In the end, after we redefine the superfields X and T to be dimensionless and canonical, we have and where we have defined Then, if we assume that the supersymmetry breaking scale of the Volkov-Akulov model serves also as the UV scale where the nilpotency of X is imposed, we would have If, instead, we consider the Volkov-Akulov model to be a low energy effective description of some supersymmetric model with supersymmetry breaking scale √ f , then we might wish to impose the boundary conditions (2.11) at some lower scale Λ. This is captured by choosing another value for ξ U V , i.e. we would have We should note immediately that the qualitative results regarding the vacuum stability will not depend on ξ U V as long as ξ U V ≥ 1. In fact, the tachyonic behaviour that we will shortly demonstrate will only become more extreme as ξ U V increases. For this reason, we will use ξ U V = 1 in our numerical examples, as the maximally benign option.
We should also observe that, here, we consider the pure Volkov-Akulov model, which we might imagine as the low energy limit of a supersymmetry breaking model where all other degrees of freedom are sufficiently massive and can be integrated out. More generally, it could be possible that some light degrees of freedom remain in the effective theory below Λ, and could have non-trivial couplings to the nilpotent superfield X. In this case, one would have to include the effects of these couplings on the RG flow. However, it is worth noting that the superfield T starts out as a Lagrange multiplier that does not couple to any other degrees of freedom except X. This means that at least for small t the presence of additional light degrees of freedom in the EFT would not be able to greatly affect the evolution of β and γ. Thus, we expect the qualitative features of the results described in the next section to remain valid even in the context of more general models.
Finally, as a non-trivial cross-check of our results, we can study the flow of the coupling q with the use of the fermionic terms that are related to the relevant part of the Kähler potential, which are given by after Wick rotation, but still in the dimensionful notation. The term in (2.60) is influenced by the fermionic propagator and one needs the ERG equation for such fields. We havė where λ is the fermion component of the superfield T , which is explicitly defined as In order to find the flow of q we need the superpotential part that enters L int. , namely, in terms of dimensionless fields, The right hand side of the ERG equation is then where we have abbreviatedlα α = σ mααl m in the middle lines. The left hand side of (2.61), always in terms of dimensionless fields and couplings, is theṅ so that we can deduce the equationζ

What are the bound states?
Here we wish to determine the nature of the composite states described by X and T . The scalar X is clearly a multi-linear bound state of goldstini, which is controlled by G 2 /2F X , which contains a goldstino bilinear, taking into account that F X has an on-shell expansion in terms of the goldstino.
The scalar T is also multi-linear in the goldstini in the on-shell Volkov-Akulov theory. To see this let us first take the superspace equations of motion for X, before imposing the nilpotency condition via T , which give On this superspace equation one can then impose the condition X 2 = 0, because it is derived from the independent variation of T . From (2.68) we see that, once we project to the lowest components, we have T X = −f + D 2 X|/4 = −f − F X and, using the on-shell value of F X , we find 3 Consequences for the pure Volkov-Akulov model

Critical point stability analysis
To analyze the properties of the model at a lower energy scale, we refer to (2.55) and (2.56) and use the regulator c(p 2 ) = (1 −p 2 )Θ(1 −p 2 ), which gives c 1 = −1 and N = − 1 32π 2 . We stress once again that we can make this choice without loss of generality as far as the lower energy dynamics are concerned. The Kähler potential and the superpotential become where the couplingsζ = ζ 2α √ β are obtained by canonically normalizing the fields, i.e. dividing by appropriate powers of the wavefunction renormalization.
The scalar potential is defined as where the indices i, j, ... = 1, 2 run over the complex scalar fields X and T and g ij is the inverse of the scalar field space metric g ij = ∂ i ∂ j K.
Once X and T (and their complex conjugates) are expressed in terms of real scalar fields via it can be easily shown that the scalar potential has a critical point at where its value is .
This is a positive energy configuration, where supersymmetry is spontaneously broken.
The (in)stability of such critical point is deduced from the scalar mass matrix associated with V , evaluated on the configuration itself. The (multiplicity two) eigenvalues are As it can be clearly seen, at least one of the eigenvalues (3.7) is negative: the scalar field space origin is always tachyonic. We can further observe that, as the RG time flows, the term g/f 2 decreases, rendering all the eigenvalues negative for sufficiently large t (t 0.35 for ξ U V = 1), providedγ andζ are positive, which is the case. Choosing ξ U V larger will simply increasef and make the tachyonic behaviour more extreme.
The investigation of the existence of other critical points and the possible consequent discussion of their relevance can be simplified by exploiting the following observation. The Kähler potential and the superpotential have a R-symmetry under which the superfields X and T have opposite non-vanishing R-charges. This means that the scalar potential will have a R-symmetry, even though the latter may or may not be preserved by all the other (e.g. higher derivative) interactions. Therefore, once we leave the central configuration, one of the scalar fields is bound to behave like a R-axion Goldstone mode, at least as far as the scalar potential is concerned.
Then, by definition, such a mode will be massless and it will have a shift symmetry on the critical points that are away from the central one. Therefore, we can set it to vanish without loss of generality. We can consistently choose σ, which is bound to contribute to the R-axion as long as τ = 0, to be With this choice, we see that in order for the gradient of V to be able to vanish, χ has to be set to zero too. We can thus restrict to the φ and τ directions to search for other possible critical points. As long as a positive energy critical point, which is also tachyonic, can be found. Moreover, for larger t, the characteristic field values for this critical configuration are φ 3 ∼f /g and τ 3 ∼f 2 /g 2 . Sincef grows exponentially with t, coherently with the small field approximation we are working with, these critical points become untrustable very soon (already for t ∼ 1) along the RG flow.
To give a flavour of the behaviour of the scalar potential V along the non-axionic directions φ and τ , we include in Figure 1 the stream plots of the (opposite of the) potential gradient at t = 0.1 and t = 1, for ξ U V = 1, where one can observe the appearance of the tachyons.

Why is there a tachyon in the central critical point?
We will now argue that the existence of the central tachyon that we have just encountered is unavoidable for a consistent RG flow.
First, let us note that the superpotential at the UV point (identified by an energy scale, say, masses to the scalars X and T has been generated during the flow, then, because the effective masses typically increase as RG time passes, there would be a scale Λ 1 below which we can remove the scalars T and X and work with a new set of constrained superfields satisfying µ ≤ Λ 1 < Λ 0 : X 2 = 0 = XT . Let us note that, in contrast to the scalar masses, the masses of the fermions are protected by R-symmetry: therefore, they remain zero. Indeed, the R-symmetry assignments are and there is no mass combination that can yield a R-invariant. The only way in which a scalar could remain massless is if it was a R-axion, but on the central critical point the R-symmetry is unbroken. Let us repeat the same procedure by decreasing the energy scale µ below Λ 1 and find the new effective theory. Initially, the superpotential takes the form 13) and the Kähler potential is making explicit the role of Y and Φ as Lagrange multipliers. This being fixed, we lower the energy below Λ 1 and characterize the resulting effective theory. We can again observe that the superpotential has the appropriate vertices to generate the kinetic terms for the superfields Y and Φ and we further notice that, because of the Yukawa couplings, the condition (3.10) is still valid on the central critical point, which is now located at T = 0 = X = Y = Φ. As before, on this configuration the fermions are massless by virtue of the R-symmetry and the scalars are bound to become heavy as we go to lower energies. We can consequently integrate out once more all the heavy scalars below some energy scale Λ 2 defining a new effective theory characterized by µ ≤ Λ 2 < Λ 1 : The problem is then manifest. Unless there is a dynamical reason to stop this procedure, we could get infinite new states in the deep IR, which lead to a possible series of inconsistencies. Therefore, the flow self-consistently terminates itself by introducing tachyons that, once appearing, can not be decoupled in a consistent way and, as a consequence, this domino effect stops.

Limitations of the SLPA
The results that we have obtained above have all been derived in our supersymmetric rendition of the local potential approximation (SLPA), which ignores the generation and feedback of higher derivative terms in the exact renormalization group equations. The LPA is a well-motivated and tested approximation and it is a common practice in ERG calculations, while the SLPA is a minimal modification of it, motivated by supersymmetry. However, it is important to discuss its regime of validity and the possible corrections that one could expect to our results because of its use.
Since we start from a UV model that has vanishing non-Kähler interactions, the higher derivative terms have to be generated before feeding back into the flow for the scalar potential and their effect is expected to appear at higher order in the RG time. The SLPA can be regarded as accurately giving the RG flow for a small decrease in the energy scale. A way to see this consists of solving the full exact renormalization group recursively by discretizing t and starting from the UV values of the couplings, that is by making reference to the Kähler potential K = |X| 2 and the standard superpotential (2.8). In addition, for concreteness, we will consider the flow with the use of the optimized regulator c(p 2 ) = 1 −p 2 Θ 1 −p 2 and we will keep the propagator pieces for both T and X. In the first step one would generate the SLPA terms d 4 θ|X| 4 and d 4 θ|X| 2 |T | 2 and, on top of them, the higher derivative term d 4 θT ∂ 2 T , which is quadratic in the auxiliary fields, together with the higher derivative term d 2 θX 2 ∂ 2 T , which is linear in the auxiliary fields. Such higher derivative terms are ignored in the SLPA.
The next recursive step would immediately give the wavefunction renormalization of X and A similar argument can be made regarding the dynamic nature of T . Given that at small t, where the SLPA can be trusted, the γ coupling is positive, T clearly acquires a positive kinetic term. Once this happens, it is impossible for the flow to bring this kinetic term back down for any finite energy, because it would require significant effects from higher derivative terms, breaking the EFT description. Moreover, if the corrections managed to pull the kinetic term of T back to zero at any finite energy, at energies below that we would run the risk of obtaining ghosts.
A possibility that is harder to rule out is that additional couplings appear due to the effects of higher derivative terms that manage to stabilize the potential away from the central critical point. The SLPA results already indicate the presence of additional (still tachyonic) critical points and additional terms, which could arise from higher derivative contributions, could help in stabilizing them. For instance, a |T | 4 term in the Kähler potential can be generated via higher derivative terms with its coupling constant parametrically suppressed relative to ζ and γ. We stress that, even in this case, the conclusion that a goldstino condensate forms remains valid and the dynamics of the theory around this new vacuum will need to be re-examined.

Coupling to pure supergravity
Let us now briefly discuss the supergravity embedding of the model that we have so far presented in the framework of global supersymmetry. As a preliminary important comment, we would like to emphasize that the impact of the quantum effects that are related to the supergravity sector is not taken into account here. In other words, we are simply considering the Kähler potential and the superpotential (with a possible addition of a constant term) of the composite supersymmetric theory, namely (2.55) and (2.56) (or, more specifically, (3.1) and (3.2)), coupled to classical supergravity. If the composite fields take values that are parametrically smaller than the cut-off, then the following analysis can be trusted as a first approximation.
Since we are accessing Supergravity and, therefore, the Planck mass enters as an additional energy scale, we will deal with it in following way. We can set where a realistic value for the dimensionless parameter P could be, for instance, P 10 4 . After writing all expressions in terms of dimensionless fields, couplings and momenta, this translates to replacing every instance of M P by e t P , which is the value of the Planck mass in units of µ. The exponential e t is actually the "classical" flow of M P , simply because it has dimension [M P ] = 1, as it is the case for any dimensionful coupling that does not flow due to the leading quantum effects that we investigate here.
As mentioned above, we also slightly modify the superpotential by introducing a constant term, which is related to the Lagrangian gravitino mass for a Kähler potential K and a superpotential W .
In accordance with our conventions, we write the superpotential as Typically, the superpotential constant term is chosen to be independent of M P , so that the gravity decoupling limit M P → +∞ is captured by m 3/2 approaching 0; here, however, we measure it directly in Planck units: therefore, [W 0 ] = 0.
We then make use of (3.1) and (4.3) to calculate the scalar potential where D i W is the Kähler covariant derivative of W , If and only if W 0 = 0, (4.4) has a de Sitter critical point at Computing the scalar mass matrix at such critical configuration, we find that, as in the rigid case, it is highly tachyonic with masses similar to (3.7), in accordance with the refined de Sitter conjecture [8][9][10][11]. If W 0 is small, the critical point moves away from the origin (X,T ) = (0, 0) and it still remains highly unstable. Meanwhile, for large enough W 0 , the potential remains tachyonic, but also gets pulled down to negative energy.
It is also worth noting that our conclusions agree with other results in the literature, and in particular with [38,44], where a tachyon shows up in the central critical point. Here, however, we obtain these results in a manifestly supersymmetric setup.

Consequences for anti-brane uplifts
In this section we explore some consequences of the new composite state dynamics and their RG flow for string theory constructions involving anti-brane uplifts of AdS vacua to meta-stable de Sitter critical points. The effect of the anti-brane in these constructions is meant to be captured by adding the Volkov-Akulov Kähler potential and superpotential to those of the supergravity model describing the pre-uplift system. In this way, we assume that the Volkov-Akulov system couples to the other ingredients only via supergravity interactions and the scalar potential for the fields X and T , as well as their RG flow, should only be affected by Planck suppressed corrections 7 . With these assumptions, we can spell out at least two important consequences for models involving nilpotent chiral multiplets.
First, we should expect the tachyonic behaviour near the origin of the (X,T ) field space to remain (and we will verify this explicitly within the KKLT setup). The endpoint of this instability will depend on the specifics of additional non-renormalizable operators in the theory. In principle, one expects such corrections to appear from String Theory as well as from corrections to the local potential approximation. This may stabilize the system, but the final configuration is likely to lie at large values of X and T and its physical interpretation is therefore unclear and deserves further investigation.
Second, regardless of the location of the final configuration, one can ask whether it has any chance of remaining at positive energy values. In models with anti-brane uplifts, we note that the final vacuum energy is typically the result of a competition between two dominant terms in the scalar potential with other contributions being M P -suppressed. Here, f is the coefficient of the linear term in X in the Volkov-Akulov superpotential, while V 0 is the pre-uplift contribution to the energy that is independent of the fields X and T , but may depend on the other fields in the model.
Incorporating the RG flow and working in terms of dimensionless and canonically normalized fields as in the previous sections, the f 2 term will flow as where the exponential behaviour corresponds to the "classical" RG flow and the 1/α(t) factor results from the wavefunction renormalization of the field X. The V 0 term will, of course, also have the "classical" exponential growth; however, it will not inherit the wavefunction renormalization of the fields X or T . At this point we note that α(t) is a monotonically increasing function, and thus the uplift term in the potential becomes suppressed at lower energies. If the superpotential contains a constant term, as it is typical in most models of moduli stabilization, its contribution to the potential will not receive any additional suppression. This means that there is a tendency for V 0 to dominate over the uplift term at lower energies, possibly resulting in an AdS vacuum.
It therefore appears that, in order for the uplift term to remain "competitive" at lower energies, the superpotential can not contain terms that are independent of any degrees of freedom in the low energy effective theory. This also means that any heavy moduli that are integrated out must not have VEVs contributing to the superpotential.
Note that in our approach α(t) grows linearly at large t. This behaviour, however, will be corrected by the anomalous dimension of X, which our approach ignores. As a first check, we can naively insert a small anomalous dimension δ into the flow equation for α, givinġ For negative anomalous dimension, the growth at large t will be more rapid, exacerbating the problem described above, while for positive anomalous dimension the linear growth is expected to stop and approach a finite value, that is of order (γ IR + ζ IR )/δ. In this last case, maintaining positive energy might remain possible, but it requires very small V 0 .
There is another interesting possible caveat to the above argument arising from the fact that in a quasi-de Sitter state the Hubble scale provides an IR cut-off, which could potentially halt the RG flow before the negative contributions to the scalar potential overtake the uplift term.
Let us imagine a scenario where the scalar potential, evaluated for a particular value of the RG time t, has a critical point whose energy is given by an expression of the form (4.7), with the uplift term flowing as (4.8). The Hubble scale measured in units of the RG scale µ, sourced by this potential, is for large (enough) t. Consistency requires that our renormalization scale is above the apparent Hubble scale derived from this potential which can be written as with t * > t. The combination of these expressions gives The condition that t * > t can potentially put a stop to the RG flow, provided that the above equation can be satisfied when setting t = t * . For V 0 = 0 there is always a solution as long as α(t) does not grow exponentially, since the exponential on the left hand side of (4.12) overpowers the sub-exponential growth of α(t) and both sides of the equation asymptote to zero. For small enough V 0 , a solution continues to exist and pushes t * higher. In either case, the RG flow will eventually stop at a finite value of t, resulting in a de Sitter critical point with a Hubble scale that is exponentially suppressed relative to the UV cut-off.
On the other hand, for sufficiently large V 0 the large t solution to (4.12) disappears entirely,

Re[T]
-0. meaning that the IR cut-off disappears, and the effects of the anomalous dimension of X remain the only potential mechanism of staying at positive energy.
We can investigate these effects in the familiar KKLT setup. We couple our description of the Volkov-Akulov model to an additional chiral multiplet S governed by the pre-uplift KKLT Kähler potential and superpotential. This means that we have K = −3M 2 P log[(S +S)/M P ] + |X| 2 and W = W 0 + Ae −aS/M P + f X, where X 2 = 0. As mentioned above, we will assume that the additional supergravity couplings do not greatly affect the RG flow for the X and T couplings as well as only include the "classical" running for the couplings in the S sector.
In our dimensionless conventions the total Kähler potential and superpotential now take the form K = −3P 2 e 2t log S +S P e t + K norm. Sitter meta-stable minimum (see Figure 2). However, evolving the theory down the RG flow even slightly, the field T becomes dynamical and the nilpotency condition on X is relaxed. The de Sitter critical point moves slightly in the (X,T ) plane. More importantly, this critical point is not stable in the X and T directions, but develops a tachyon roughly along the T direction (see Figure 3). As in the rigid case, the tachyonic behavior appears for all values of ξ U V and only becomes worse as we increase it.
The endpoint of this tachyonic instability is unclear; however, even by considering the potential for fixed S, we can see that it rolls down to a configuration with negative energy at field values of O(1), where possible higher order terms in the Kähler potential become important (see Figure 4). The relation of the tachyonic instability that we are finding here to the "goldstino evaporation" setup of [72] or the KPV scenario [73], which end in supersymmetric points, is yet unknown.
In the above analysis we have tuned the parameters so as to produce the de Sitter critical point in the UV, where the field X is nilpotent. Following the RG flow to a lower energy scale, we find precisely the second problem described above. The linear term in X in the superpotential, which is responsible for the uplift, has the form 1/2 X (4.14) and, together with the W 0 terms, it represents the main contribution to the scalar potential: − 3P 4 e 4t |W 0 | 2 + . . . .

Conclusions
In this work we analyzed the possible composite states that can be generated from the goldstino in the Volkov-Akulov model due to the fermionic self-interaction. Following an exact renormal-

A ERG equations for chiral supermultiplets
We derive here the ERG equations for a supersymmetric model involving chiral multiplets. Note that certain conventions and normalizations differ from those of [50], which we have used in the main text. These will be pointed out when they will occur.
Let us consider the generating functional where Λ represents the UV cut-off and µ is an energy scale of interest below Λ; Φ is a collective symbol denoting all the fields of our theory, with individual fields labeled as Φ A and L[Φ; µ] is the (Euclidean) action at a given energy scale µ. where our convention for the variational derivatives is defined by, say, (with the normalization of the momentum matching δ-function including a factor of (2π) 4

A.1 Bosons
To proceed, we split the action into a propagator and an interaction piece. Namely, we write the action L[Φ; µ] as AB (p, µ) defines the propagator including a regulator function. We now restrict our attention to {Φ A } A being the bosonic fields {φ a } a or {F a } a . Since we are assuming massless propagators, we can diagonalize them through appropriate field redefinitions so that p 2 δ ab for real scalar fields φ a ; Notice that there is a factor of √ 2-difference in the normalization of the auxiliary fields compared to that used in the main text. For the real scalar and auxiliary fields this diagonalization allows us to omit the Kronecker-δ and the species index from the C (Φ) 's. The scaling dimensions of these propagators determine the (classical) dimensions of the corresponding fields, which we will denote as ∆ Φ . In momentum space they are ∆ φ = −3 and ∆ F = −2. This, in turn, determines the scaling dimensions of the various couplings in L int. . We will denote such couplings as {g λ } λ and their scaling dimensions as {∆ λ } λ .
At this point, let us observe that only the couplings and the propagator have an internal µ dependence, while the fields and the momenta are µ-independent. We wish, however, to work with dimensionless fields, couplings and momenta and so we carefully track their µ dependence.
The dimensionless quantities are defined aŝ We also define the dimensionless propagators aŝ in such a way that the bosonic action propagator term can also be written as for real scalar fields φ a ; Φ=F =Ĉ (F ) = c(p 2 ) for real auxiliary fields F a . (A.11) To compute the left hand side of (A.5), we will need the µ-derivatives of all the dimensionless quantities that we have introduced. From (A.8) we obtain −µ∂ µp = −pµ∂ µ µ −1 = pµ −1 =p (A. 12) and similarly for the momentum space measure, −µ∂ µ dp = dp. We also have −µ∂ µ c(p) = (−µ∂ µp )∂pc(p) =p∂pc(p) , where Φ A denotes a regular derivative of Φ A with respect to its argument.
Armed with these expressions we can computė C (F ) (p) = −(µ∂ µp )∂pĈ (F ) (p) =p∂pc(p) . (A.14) Defining, then,C From this we can computė so that the left hand side of (A.5) iṡ prop. +L int. , (A. 18) where in the first term a sum over all scalar and auxiliary fields is understood.
We now show that for a suitable choice of {Ψ A } A=φ,F this first term also appears on the right hand side of (A.5) and do not take part to the final ERG equation. A similar procedure, which we will describe further, allows to get rid of the fermionic propagator terms.
The appropriate choice of Ψ A [Φ; µ] is as follows: δΦ A (k) withL = −L prop. + L int. , (A. 19) andC (Φ) being defined by (A.15). Omitting the functional and function variables of the action, the right-hand side of (A.5) consequently becomes 20) where no sum over A is implied. The evaluation of the terms in the last line gives which can be absorbed into the measure, as well as which is precisely the term that we need to cancel its analogous on the other side. Putting things together, we obtain the ERG equatioṅ We can now expandL int. so that we can isolate the behaviour of individual coupling constants. As already mentioned, we will denote these couplings as {g λ } λ and their classical scaling dimensions as {∆ λ } λ , where λ is an index labelling the various interaction terms. A nonderivative interaction term will have the form

A.2 Fermions
For a single Weyl fermion we can write the propagator part of the action as where the fields are dimensionless (and are related to their dimensionful partners by ∆ χ = −5/2), but we omit the hats on them to avoid notation clutter, and we also havê partial derivatives, and they differ by a factor of (2π) 4 when it comes to momentum matching.
We make a final observation. For each massless chiral multiplet, which consists of a complex scalar field, a Weyl fermion and a complex auxiliary field, it is straightforward to check that the terms that would otherwise have to be absorbed into the measure in the right hand side of the ERG equation actually sum up to zero, which is a non trivial relation required by supersymmetry.