UV Corrections in Sgoldstino-less Inflation

We study the embedding of inflation with nilpotent multiplets in supergravity, in particular the decoupling of the sgoldstino scalar field. Instead of being imposed by hand, the nilpotency constraint on the goldstino multiplet arises in the low energy-effective theory by integrating out heavy degrees of freedom. We present explicit supergravity models in which a large but finite sgoldstino mass arises from Yukawa or gauge interactions. In both cases the inflaton potential receives two types of corrections. One is from the backreaction of the sgoldstino, the other from the heavy fields generating its mass. We show that these scale oppositely with the Volkov-Akulov cut-off scale, which makes a consistent decoupling of the sgoldstino nontrivial. Still, we identify a parameter window in which sgoldstino-less inflation can take place, up to corrections which flatten the inflaton potential.

The connection of such theories to string theory has recently been studied in [15,[22][23][24][25][26][27]. Effective supergravity theories with a constrained goldstino multiplet can be shown to arise from D3-branes in certain geometries [23][24][25][26][27]. The emergence of nonlinear supersymmetry in string models with anti-branes was proven in [28], in the context of global string theory vacua [29]. In such UV embeddings it is difficult to extract the behavior of the supergravity above the cut-off scale of the Volkov-Akulov action. Usually there is no scale at which linear supersymmetry is restored and therefore the scalar component of S does not exist. A step towards understanding the connection between the linear and nonlinear regimes was recently made in [30], where it was shown explicitly that nonlinear supergravity theories are equivalent to linear supergravities with an infinitely heavy sgoldstino scalar, and that the limit relating the two is well-defined via functional integration. With a few restrictions this connection was previously known in the rigid limit [31]. 1 Therefore it is desirable to study field theory examples in which a heavy sgoldstino exists so that supersymmetry becomes linearly realized at a high scale. In such cases, the sgoldstino field cannot be infinitely heavy. Its mass, and hence the Volkov-Akulov cut-off scale, must be lower than the Planck scale -and favorably below the Kaluza-Klein and string scales. A stronger constraint arises from unitarity which signals a perturbative breakdown of the nonlinear theory at a scale ∼ √ m 3/2 in Planck units. A UV complete theory which can describe both the linear and nonlinear regimes is bound to yield corrections which are missed by simply imposing a nilpotency constraint on the goldstino multiplet in supergravity. In this letter we compute these corrections and evaluate their effects in simple inflation models previously studied in the literature. It is our aim to prove that in a well-defined regime of the theory, corrections are under control -though in a quite constrained parameter space. For this purpose the class of models developed in [7] is particularly instructive. 2 They feature the coupling of a nilpotent stabilizer multiplet to a holomorphic function of the inflaton multiplet, giving rise to a plethora of possible potential shapes for the inflaton scalar. During inflation and in the vacuum supersymmetry is broken by the auxiliary field of the nilpotent multiplet. The setup can accommodate low-energy supersymmetry which is nontrivial given the high scale of inflation. We extend this setup to a supergravity with heavy fields in which a large mass for the sgoldstino scalar is generated dynamically. We expect our results to be relevant in many other supergravity theories with nilpotent goldstino multiplets. Thus, we hope that this work is another step towards understanding nilpotent multiplets and their role in cosmology.

II. SGOLDSTINO DECOUPLING
The success of nilpotent fields in cosmology has triggered growing interest in their field-theoretical origin. It is well-known that in spontaneously broken linear supersymmetry, the sgoldstino field acquires a large mass through the operator in the Kähler potential. In the limit c → ∞, the sgoldstino becomes infinitely heavy and the resulting theory is equivalent to nonlinearly realized supersymmetry with a nilpotent goldstino multiplet S [30,31]. Clearly this theory is only a low-energy effective theory. With the sgoldstino decoupled, it violates perturbative unitarity at the intermediate energy scale √ m 3/2 [34]. Requiring inflation in the perturbative regime one obtains the generic constraint [7] where H denotes the Hubble scale. However, the scale of supersymmetry breaking may be different during and after inflation. Hence, nilpotent inflation models consistent with low-energy supersymmetry can be constructed [7]. A different concern is the limit c → ∞: in a UVcomplete model the operator (1) arises from couplings of S to heavy degrees of freedom. As an example, we may consider the superpotential coupling W ⊃ λSX 2 of S to the heavy field X with mass m X . This coupling generates a one-loop correction [35], The limit c → ∞ in (1) then corresponds to taking the coupling λ to infinity or the mass m X to zero. Since both must be finite and m X must be large for the effective field theory (EFT) to make sense, we must consider the regime where the sgoldstino has a finite mass, i.e., finite c. In the remainder of this letter we strive to determine whether inflation is still possible in this case. Specifically, we determine whether the inflaton potential obtained in the nilpotent limit still holds and corrections are under control.
We will find that such corrections are of two different natures. Additional heavy fields at the energy scale Λ backreact on the inflaton potential, introducing corrections which vanish as Λ → ∞. 3 On the other hand, the finite mass of the sgoldstino field leads to corrections which vanish in the limit where the latter is infinitely heavy. This corresponds to Λ → 0. Therefore, it is far from obvious that both types of corrections can be suppressed simultaneously.
Note that while we study this in the class of inflation models proposed in [7], our findings can straightforwardly be applied to alternative scenarios with nilpotent multiplets.

III. SGOLDSTINO-LESS MODELS OF INFLATION
Let us briefly review the inflation models of [7]. They feature the Kähler and superpotential where Φ denotes the inflaton superfield and S contains the stabilizer field. This setup is a generalization of the models developed in [36,37] with built-in supersymmetry breaking by the auxiliary field of S. The function f satisfies f (0) = 0, f (0) = 0 and f (x) = f (−x). In [7] it is assumed that S fulfills the boundary condition S 2 = 0 of a nilpotent chiral multiplet. This implies s = 0 for its scalar component. 4 The factor √ 3 ensures the cancellation of the cosmological constant in the vacuum at φ = 0. Along the inflationary trajectory the potential reads where ϕ = √ 2 Im φ denotes the canonically normalized inflaton. Two examples for f are discussed in [7]. One is leading to the potential of chaotic inflation, V = 1 2 m 2 ϕ 2 . The other is producing the plateau potential In the following we call S the goldstino multiplet and s the sgoldstino, its scalar component. This is because s is the heavy scalar that is supposed to decouple, and despite the fact that the inflaton multiplet has a sub-dominant but nonvanishing auxiliary field during inflation.

IV. CORRECTIONS FROM THE SGOLDSTINO
Let us discuss corrections to the inflaton potential which arise if the sgoldstino has a finite mass. To this end we consider The difference compared to the previous section is that we do not impose the nilpotency constraint S 2 = 0. Instead we introduce the term |S| 4 /Λ 2 in the Kähler potential which generates a large -but finite -mass for the sgoldstino s and dynamically keeps s close to the origin. 6 Supersymmetry breaking introduces an inflaton-dependent linear term for the stabilizer field which slightly shifts it away from the origin [4]. As this effect scales inversely with the mass of s, it is absent in the nilpotent limit. Notice that we introduced the parameter δ which allows us to tune the vacuum energy to zero at the minimum of the potential. Due to the shift of s, δ is close to but not exactly √ 3. We find For a compact notation we introduce 5 As pointed out in Section 5 of [7], the function f can be extended to include matter fields like an MSSM sector. Tachyonic directions are avoided automatically for matter fields which appear at least quadratically in f . 6 Compared to Section II we absorbed the parameter c in the definition of Λ.
denoting the inflaton potential in the limit where S is nilpotent and hence s is infinitely heavy. The gravitino mass along the inflationary trajectory can be approximated as As only the real part of the stabilizer field is displaced during inflation, we sets = s in the following. At second order in s the scalar potential reads including only terms up to O(Λ 2 ). 7 The sgoldstino mass is given by which, through m 3/2 , depends on ϕ during inflation. The inflaton-dependent minimum of s lies at The scalar potential after integrating out s reads As mentioned above, the corrections from the sgoldstino sector appear in powers of m 2 3/2 /m 2 s and H 2 /m 2 s , where H ∼ √ V 0 again denotes the Hubble parameter. Corrections are under control as long as m 3/2 > HΛ which is the case during inflation in the two examples of Section III. 8 Note that even when the corrections are small the sgoldstino can affect post-inflationary cosmology. If the above constraint is violated after inflation, s may no longer trace its minimum. If it gets trapped the associated potential energy can alter late-time cosmology. This is not necessarily problematic and may even induce interesting signatures. We merely point out that decoupling s from all dynamics in the universe requires the bound m 3/2 > HΛ to be satisfied during the entire cosmological evolution. We will show in the following that in a consistent EFT Λ cannot be arbitrarily small, making this a very severe constraint. 7 Notice that s = O(Λ 2 ) and m 2 s = O(Λ −2 ). 8 For the sgoldstino to be heavier than Re(Φ) one would additionally have to require |f (Φ)|, |f (Φ)| < |f (Φ)| on the inflationary trajectory. These conditions are, however, already fulfilled by requiring slow-roll inflation.

V. CORRECTIONS FROM UV COMPLETIONS
In the previous section we have included corrections to the inflaton potential which arise from the sgoldstino sector. The corrections disappear in the limit Λ → 0 in which the sgoldstino becomes infinitely heavy. But there are more corrections related to the heavy fields living at the scale Λ. Contrary to the sgoldstino corrections, these scale with Λ −1 and prevent us from taking the limit Λ → 0. In the following we discuss these UV corrections in two examples. In the first example the sgoldstino mass is generated by Yukawa interactions with heavy fields, in the second example by gauge interactions. Despite their simplicity we expect that our examples are representative of more sophisticated UV embeddings.

A. Example 1
Consider two additional chiral multiplets X, Y with a vector-like mass M . We consider M to be large compared to the Hubble scale and the gravitino mass during inflation. We define the model as follows, It bears resemblance to the O'Raifeartaigh model [38]. The sgoldstino superfield S obtains a mass term through its coupling to X. The parameter δ is chosen such that the vacuum energy vanishes at the minimum of the potential The tree-level scalar potential along the direction x = y = 0 reads where V 0 = |f | 2 and m 2 3/2 |f | 2 as before ands = s is assumed. The imaginary part of s is stabilized at the origin and does not play a role in our discussion. The tree-level mass of s is negligible compared to the oneloop contribution due to the interaction with X. We use the Coleman-Weinberg formula where Str M 4 = i (−1) 2Ji (2J i + 1)m 4 i is the trace over the field-dependent mass eigenvalues of states with spin J i . The Coleman-Weinberg potential gives rise to an additional mass term  Note that it is the same as the mass term arising from the equivalent quantum correction to the Kähler potential We conclude that we obtain the model of Section IV as a low-energy effective theory and the small shift of s does not affect inflation for sufficiently small Λ. However, we have yet to consider the effect of inflation on the sector of heavy fields X and Y . Inflation does not induce linear terms for the scalar components x and y. However, it generates a bilinear mass term for x. The mass of Im x is given by Thus, taking the limit M → 0 to make S nilpotent introduces a tachyonic direction in the full theory, which makes inflation impossible. 9 To obtain a positive squared mass, we obtain the constraint Taking the example of chaotic inflation (6) and using m 6 · 10 −6 , ϕ ∼ 15, this translates into M > 0.03 √ λ. At the same time, to make the sgoldstino sufficiently heavy we have to require that Λ 1 which is equivalent to M 0.09λ 2 , cf. (22). After combining these two constraints there is a small window at λ 1 and M ∼ 0.05, where sgoldstino-less inflation can consistently take place. In this regime the heavy fields remain at their minima and inflation does not receive corrections besides those of Section IV. Notice, however, that the sgoldstino mass can at most be enhanced by an O(10) factor compared to the gravitino mass. This is illustrated in Table I, where we show a possible choice of scales which leads to successful sgoldstino-less inflation. In the vacuum, the gravitino mass is much smaller than in the inflationary epoch and low-energy supersymmetry can be obtained.

B. Example 2
Second, we consider an example where the sgoldstino receives its mass from gauge interactions. We introduce three new chiral multiplets X, Y , Ψ which carry the charges q(X) = −1, q(Y ) = 1 and q(Ψ) = 0 under a U(1) symmetry. 10 We further assume that q(S) = 1 and define the model by where δ is again chosen to adjust the cosmological constant. We find The second term in the superpotential is introduced to break the U(1) symmetry at a high scale. For the same reason as before we setx = x,ȳ = y,ψ = ψ in the following. The imaginary parts do not play a role in our discussion. Given that v Max(m 3/2 , H) the U(1) symmetry is broken along the almost D-and F -flat direction Using these three conditions to eliminate x, y, and ψ yields the scalar potential with This resembles (12) if we identify Λ = 4/3v. 11 The large mass m s decouples the sgoldstino and the small shift of s does virtually not affect inflation. Unfortunately, this is not the end of the story. So far we have worked in the regime m 3/2 , H v. We expect additional corrections if either m 3/2 or H are close to the scale v. To find these corrections we must treat x, y and ψ as dynamical fields. We perform a Taylor expansion around s = 0, ψ = 0, x = v, y = v up to second order in the shift of the four fields. Setting the four fields to their new minima, we arrive at the following effective inflaton potential 10 In order to avoid anomalies we have to introduce another field Z with charge q(Z) = −1. The field Z can be coupled to a new singlet Θ via a term Y ZΘ in the superpotential. When Y breaks the U(1) symmetry this becomes a large vector-like mass term for Z. In this case Z and Θ do not affect our analysis, and we neglect them in the following discussion. 11 To recover the exact form of (12) we would have to substitute Notice the difference to our first example. In this case the shift of the heavy fields during inflation causes a backreaction on the potential. Expression (31) only includes the corrections due to the heavy fields. In addition, the sgoldstino-induced corrections of Section IV arise.
Requiring the correction to be suppressed compared to the leading-order inflaton potential leads to the constraint v m 3/2 for λ, g ∼ O(1). In the model of chaotic inflation defined by (6), with m ∼ 6 · 10 −6 and ϕ ∼ 15, the constraint translates into v 0.03 .
Even for larger v there are substantial corrections. We depict the effective inflaton potential of the example (6) in Figure 1 for f 0 = 10 −14 , m = 6 · 10 −6 , λ = g = 1, and different values of v. Again there is a small window at v 0.1 where the backreaction is under control and sgoldstino-less inflation can take place. As in the previous example, choosing v too large decreases the mass of the sgoldstino scalar beyond the point where it can be consistently decoupled. The corrections from the heavy fields of the UV completion cause a flattening of the inflaton potential.

VI. DISCUSSION
We have emphasized that sgoldstino decoupling in cosmology is nontrivial. Working in spontaneously broken linear supergravity, instead of imposing a nilpotency constraint by hand we assumed that the mass of the sgoldstino field is produced dynamically. This required the inclusion of heavy degrees of freedom which couple to the sgoldstino. We discussed two possible UV embeddings of nilpotent goldstino multiplets. Both scenarios result in the sgoldstino-less inflation models of [7] as a low-energy effective theory. The sgoldstino has a large but finite mass m s ∼ m 3/2 /Λ during inflation, where Λ is the mass scale of the heavy fields which couple to the sgoldstino. As m 3/2 > H the sgoldstino decouples from the inflationary dynamics.
The scale Λ sets a new cut-off at which the low-energy effective theory breaks down and the heavy fields become dynamical degrees of freedom. For inflation to take place in a controlled regime, where the heavy fields can be integrated out, one has to require that the Hubble scale does not exceed Λ. However, an even more severe constraint arises in the class of models [7] which feature m 3/2 H during inflation. There inflation induces large "soft terms" which may destabilize the heavy fields. Depending on the specific UV embedding, we find that tadpole terms ∝ m 2 3/2 M 2 P /Λ and bilinear terms ∝ m 3/2 M P are particularly dangerous. In all examples we find that inflation is generically spoiled if Λ 0.1. This does not leave much room for a complete theory below the Planck scale with a decoupled sgoldstino. Still, a window of viable parameter choices survives in which sgoldstino-less inflation can successfully take place and the backreaction on the heavy fields in under control. Within this window we calculated the corrections to the inflaton potential which typically appear in the form of flattening effects.
The constraints on Λ imply that the sgoldstino mass can at most be enhanced by one order of magnitude compared to the gravitino mass. Requiring the sgoldstino to decouple in the post-inflationary cosmology as well puts strong additional constraints on the form of the scalar potential.
Due to the structure of the dangerous terms that arise, we expect these results to be relevant for many other applications of constrained multiplets in cosmology.