Cosmology with orthogonal nilpotent superfields

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Citation Ferrara, Sergio, Renata Kallosh, and Jesse Thaler. "Cosmology with orthogonal nilpotent superfields." Phys. Rev. D 93, 043516 (February 2016).

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Detailed Terms PHYSICAL REVIEW D 93, 043516 (2016) Cosmology with orthogonal nilpotent superfields

† ‡ ,1,2,3,* Renata Kallosh,4, and Jesse Thaler5, 1Theoretical Physics Department, CERN CH1211 Geneva 23, Switzerland 2INFN–Laboratori Nazionali di Frascati Via Enrico Fermi 40, I-00044 Frascati, Italy 3Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA 4SITP and Department of Physics, , Stanford, California 94305, USA 5Center for , Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 8 December 2015; published 9 February 2016) We study the application of a supersymmetric model with two constrained supermultiplets to inflationary cosmology. The first superfield S is a stabilizer chiral superfield satisfying a nilpotency condition of degree 2, S2 ¼ 0. The second superfield Φ is the inflaton chiral superfield, which can be B ≡ 1 ðΦ − Φ¯ Þ B S SB ¼ 0 combined into a real superfield 2i . The real superfield is orthogonal to , , and satisfies a nilpotency condition of degree 3, B3 ¼ 0. We show that these constraints remove from the spectrum the complex scalar sgoldstino, the real scalar inflaton partner (i.e. the “sinflaton”), and the fermionic inflatino. The corresponding model with de Sitter vacua describes a , a massive gravitino, and one real scalar inflaton, with both the goldstino and inflatino being absent in unitary gauge. We also discuss relaxed superfield constraints where S2 ¼ 0 and SΦ¯ is chiral, which removes the sgoldstino and inflatino, but leaves the sinflaton in the spectrum. The cosmological model building in both of these inflatino-less models offers some advantages over existing constructions.

DOI: 10.1103/PhysRevD.93.043516

I. INTRODUCTION also be realized nonlinearly. In such cases, the corresponding goldstino multiplet contains a spin-1=2 fermion, but it does De Sitter space plays a crucial role in our current under- not contain a spin-0 or spin-1 partner; rather, a nonlinear standing of cosmological observations [1]. The inflationary epoch in the early universe occurs in approximate de Sitter operation flips a one-particle fermion state space, and the present day acceleration of the universe will into a two-particle fermion state. It was recognized a long lead to de Sitter space asymptotically. If supersymmetry is time ago in [4] that nonlinear realizations of supersymmetry realized in nature, then it must be spontaneously broken can also be described using constrained superfields, and during any de Sitter phase. It is therefore interesting to study interest in constrained multiplets was renewed in [5,6]. de Sitter vacua with spontaneously broken local supersym- While constrained multiplets were originally introduced metry, especially in cases where the interaction of matter for global supersymmetry, they can be consistently gener- with is an essential ingredient. alized to local supersymmetry, as will be explained below. In a cosmological context, one can ask whether super- In this paper, we show how single-field can be symmetry is linearly or nonlinearly realized in a de Sitter consistently embedded in local supergravity with the help of phase. Standard linear multiplets involve partner particles constrained multiplets. The resulting supergravity action is which differ by a half-unit of spin [2]: spin-0 scalars with surprisingly economical, since it describes the dynamics of spin-1=2 fermions, spin-1=2 fermions with spin-1 vectors, just a single real inflaton, a graviton, and a massive gravitino. spin-3=2 gravitino with spin-2 graviton, and so on. These Our construction uses the superfield content of [7],withtwo supermultiplets form representations of the superalgebra constrained chiral multiplets S and Φ. The stabilizer field S 1 and the supersymmetry operation flips the spin of the state satisfies a nilpotency condition of degree 2, by 1=2. It was first recognized in [3] by Volkov-Akulov S2 ¼ 0 ð Þ (VA) that spontaneously broken global supersymmetry can ; 1:1 and the inflaton is embedded in an orthogonal multiplet Φ * † satisfying [email protected][email protected] 1We use the terms “nilpotent” and degree of nilpotency as if Published by the American Physical Society under the terms of superfields were square matrices: a superfield Xðx; θ; θ¯Þ is the Creative Commons Attribution 3.0 License. Further distri- nilpotent of degree r if r is the least positive integer such that bution of this work must maintain attribution to the author(s) and Xr ¼ 0. The term “orthogonal” will be used here as in the case of the published article’s title, journal citation, and DOI. vectors: two superfields X and Y are orthogonal if XY ¼ 0.

2470-0010=2016=93(4)=043516(15) 043516-1 Published by the American Physical Society FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016) 1 constant in local supersymmetry, the S multiplet always SB ¼ 0; B ≡ ðΦ − Φ¯ Þ: ð1:2Þ 2i contributes a positive , so it naturally appears in studies of de Sitter space. The minimal case of The real superfield B also satisfies a nilpotency condition of local supersymmetry coupled just to S leads to a nonlinear degree 3, B3 ¼ 0 [5]. Naively, a chiral inflaton muliplet Φ realization of pure de Sitter supergravity without low would contain a fermionic inflatino partner and an additional “ ” 2 energy scalars [23,24]. The action is complete with all real sinflaton scalar partner, but such states are absent in higher order fermion couplings, and this complete locally the nonlinear realization. As we will explain, this presents supersymmetric action can be subsequently coupled to new opportunities for inflationary model building, since matter fields [24–26]. In theory, nilpotent multiplets cosmological challenges presented by the inflatino and arise [27,28] when constructing supersymmetric versions sinflaton are now absent. of Kachru-Kallosh-Linde-Trivedi (KKLT) uplifting to de S2 ¼ 0 The importance of the constraint for cosmology Sitter vacua [29].4 The nilpotent multiplet plays an impor- S is already well known. The unconstrained linear multiplet tant role in KKLT and large volume scenario moduli has components stabilization scenarios, in particular for applications to particle phenomenology and cosmology [33]. Nilpotent S ¼ðS; χs;FsÞ; ð1:3Þ multiplets also appear in studies of supersymmetry break- where χs is the goldstino, S is its scalar partner (i.e. the ing with multiple “goldstini” [34]. In our view, the fact that sgoldstino), and the Fs auxiliary field is the order parameter the complete VA nonlinear goldstino action appears on the for supersymmetry breaking. While there have been D- world-volume in [27] suggests that attempts to identify the sgoldstino with the inflaton [8], the consideration of constrained versus unconstrained sgoldstino inflation scenarios face a number of challenges multiplets (and linear versus nonlinear realizations) is [9]. Therefore, it is typically necessary to stabilize S and broader issue than originally envisaged. decouple it from cosmological evolution. For example, in The goal of this paper is apply the logic of nonlinear realizations and superfield constraints to the inflaton itself. the case of a supergravity version of the Starobinsky model, Φ one can use a linear realization of supersymmetry [10] and The unconstrained inflaton multiplet has components ¯ 2 stabilize the sgoldstino via a Kähler potential term −cðSSÞ Φ ¼ðφ þ ib; χϕ;FϕÞ; ð1:4Þ [11]. More economically and elegantly, though, one can use where φ is the inflaton, b is the sinflaton, χϕ is the inflatino, a nonlinear realization of supersymmetry and simply ϕ remove the sgoldstino via the S2 ¼ 0 constraint [12] and and F is an auxiliary field. Just as the nilpotency S2 ¼ 0 in this way one can build the bosonic part of Volkov- constraint projects out the sgoldstino S and replaces Akulov-Starobinsky supergravity.3 it with a goldstino bilinear term, the orthogonality con- SB ¼ 0 χϕ ϕ A proposal to use the nilpotent multiplet in a general straint implies that b, , and F are no longer independent fields and are instead functionals of χs, Fs, and class of cosmological models was made in [13], starting φ with Lagrange multipliers in the superconformal models . The fact that superfield constraints reduce the total underlying supergravity. The stabilizer field S with S2 ¼ 0 number of physical low energy degrees of freedom was one of the motivations for the study of such constrained is now known as a nilpotent multiplet, and the resulting systems. The relevance of the constrained S and Φ fields inflationary models are known as “sgoldstino-less” models. for cosmological applications was described in [7] in the By now, there are various cosmological inflationary models context of building a minimal supersymmetric EFT for consistent with the data and yielding supersymmetry – fluctuations about a fixed inflationary background. Here, breaking at the minimum of the potential [14 17]. Many we are interested in studying the full inflationary dynamics, recent developments on sgoldstino-less models were including the end of inflation. described in [18,19], as well as other directions of work For cosmological applications, the key feature of these with constrained superfields. There are also supersymmet- orthogonal nilpotent superfields is that they have a particu- ric effective field theories (EFTs) of inflation [20,21] built larly simple form when χs ¼ 0. In a locally supersymmetric using nilpotent multiplets [7,22]. action, one can make a choice of unitary gauge for the S2 ¼ 0 It is important to mention that the nilpotency gravitino where χs ¼ 0. We will show that in this gauge condition is of more general interest beyond inflationary s ϕ ϕ cosmology. Unlike the gravitino mass parameter m3=2 χ ¼ 0 ⇒ S ¼ b ¼ χ ¼ F ¼ 0; ð1:5Þ which contributes universally to a negative cosmological such that the sgoldstino, sinflaton, inflatino, and inflaton auxiliary field are not just constrained, but entirely absent 2We use the axion-saxion naming convention here. Together, the real inflation and real sinflation form a complex scalar which appears in the lowest component of Φ. 4The idea that open string theory and spontaneous symmetry 3In [12], supersymmetry is restored at the minimum of the breaking may be associated with and nonlinearly realized potential, though, and the constrained goldstino multiplet S supersymmetry were discussed a while ago [30–32], but only becomes singular. recently were explicit constructions presented.

043516-2 COSMOLOGY WITH ORTHOGONAL NILPOTENT SUPERFIELDS PHYSICAL REVIEW D 93, 043516 (2016) from the action. This presents an extraordinary opportunity leads to three constraint equations involving the complex for cosmology. First, in cosmological models with two scalar sgoldstino SðxÞ, the fermionic field goldstino χsðxÞ,5 unconstrained superfields, one has to work hard to stabilize and the auxiliary field FsðxÞ: three of the scalars—ReðSÞ,ImðSÞ,andb—since cosmo- φ s 2 logical data favors a single scalar field inflaton .The 2 ðχ Þ S2 ¼ 0 SB ¼ 0 S ¼ 0;Sχs ¼ 0;S¼ : ð2:3Þ constraints , automatically project out these 2Fs unwanted scalar modes while maintaining a nonlinear realization of supersymmetry. Second, at the end of inflation Because the sgoldstino has been removed from the low ϕ in the presence of two chiral fermions χs and χ ,thereis energy spectrum, this nilpotent superfield has proved to be a problem of gravitino-inflatino mixing [35–37],which very useful in constructing viable cosmological models in makes the study of matter creation in the early universe the framework of two-superfield models [14–19]. very complicated. Supergravity models based on orthogonal Following [7], we introduce a second chiral multiplet Φ, nilpotent superfields have no inflatino in unitary gauge and pffiffiffi therefore no gravitino-inflatino mixing. These are obvious Φðyμ; θÞ¼ΦðyÞþ 2θχϕðyÞþθ2FϕðyÞ; ð2:4Þ advantages for cosmology. ϕ ¼ 0 In addition, the fact that F means the scalar with potential of these models is different from ones studied in the past. In the simplest case with a canonically ΦðxÞ¼φðxÞþibðxÞ: ð2:5Þ normalized inflaton, we will show that the scalar potential takes the form Here, the real scalar φðxÞ is the inflaton, the real scalar bðxÞ is its sinflaton partner, the fermion χϕ is the inflatino, and ðϕÞ¼ 2ðφÞ − 3κ2 2ðφÞ¼ 2ðφÞ − 3 2 2 ðφÞ ϕ V f g f MPlm3=2 ; F is the auxiliary field. From the antichiral superfield ð1:6Þ pffiffiffi Φ¯ ð¯μ θÞ¼Φ¯ ð¯Þþ 2θ¯χ¯ϕð¯Þþθ¯2 ¯ ϕð¯Þ ð Þ pffiffiffiffiffiffiffiffiffi y ; y y F y ; 2:6 where κ ¼ 8πG ¼ M−1, fðφÞ is related to the degree of Pl B supersymmetry breaking, and gðφÞ gives rise to a field- we can construct the real superfield , ðφÞ 0ðφÞ dependent gravitino mass m3=2 . Surprisingly, g is 1 absent from the potential, which appears to be a generic Bð θ θ¯Þ ≡ ðΦ − Φ¯ Þ ð Þ x; ; 2 : 2:7 prediction of inflatino-less constructions using constrained i multiplets. As shown in [38], this feature simplifies the This B field can be defined in a basis where the chiral construction of viable cosmological models. Here, our superfield is short, and the antichiral superfield is long. focus is showing that inflatino-less models in de Sitter Namely, we keep (2.1) and (2.4) but rewrite Φ¯ as supergravity are internally consistent. The remainder of this paper is organized as follows. We pffiffiffi Φ¯ ðyμ − 2iθσμθ¯; θ¯Þ¼Φ¯ ðyÞþ 2θ¯χ¯ϕðyÞþθ¯2F¯ ϕðyÞþ: review the structure of orthogonal nilpotent superfields in global supersymmetry in Sec. II and then generalize it to ð2:8Þ local supersymmetry in Sec. III. We highlight a counter- intuitive feature of the scalar potential in Sec. IV and We impose an orthogonality constraint on B via discuss generic aspects of inflatino-less models in V.We provide alternative inflatino-less constructions in Sec. VI Sðy; θÞBðy; θ; θ¯Þ¼0: ð2:9Þ and conclude in Sec. VII. This orthogonality relation produces a number of con- II. ORTHOGONAL NILPOTENT SUPERFIELDS: straint equations for the component fields, for each of the θmθ¯n 0 ≤ ≤ 2 0 ≤ ≤ 2 S2 ¼ 0, SB ¼ 0 with m and n . By solving these equations one finds the following [5,7]6: A. Structure in global (1) The first component of the inflaton multiplet has a We start our discussion in global superspace. Consider a real scalar field inflaton φ as well as fermionic ¯ χs chiral superfield Dα_ S ¼ 0 which is nilpotent of degree 2. In -dependent terms: the chiral basis yμ ¼ xμ þ iθσμθ¯ where pffiffiffi 5In local supersymmetry models with many matter multiplets, Sðyμ; θÞ¼SðyÞþ 2θχsðyÞþθ2FsðyÞ; ð2:1Þ the name goldstino is typically reserved for the combination v of μ various spin-1=2 fields interacting with gravitino via ψ μγ v, see χs the second degree nilpotency condition Sec. III A. The field will have this property, which justifies the name here. 6 2 Because [7] only considered fluctuations about a fixed S ðy; θÞ¼0 ð2:2Þ inflationary background, derivative terms on F were neglected.

043516-3 FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016) χs χ¯ s Φ Φ ¼ φ þ i σμ ∂ φ This can be understood since the imaginary part of 2 s ¯ s μ χs F F    contains one undifferentiated two-component spinor , s 2 s s 1 χ χ¯ μ ν χ¯ and therefore the product of three of them vanishes. þ ∂ν σ¯ σ − c:c: ∂μφ 8 Fs F¯ s F¯ s       B. Constructing an action i χs 2 χ¯s 2 χ¯s − ∂μ The most general supersymmetric action for the S and Φ 32 Fs F¯ s F¯ s   superfields at the two-derivative level is s ρ μ ν μ ν ρ χ Z Z  × ðσ¯ σ σ¯ þ σ¯ σ σ¯ Þ∂ν ∂ρφ: ð2:10Þ Fs L ¼ d4θKðS; S¯; Φ; Φ¯ Þþ d2θWðS; ΦÞþH:c: :

Note that there is no independent scalar sinflaton ð2:14Þ bðxÞ in this expression, and it has been replaced by terms bilinear or higher in the fermion χs. This is In a cosmological context, it is often useful to impose an desirable from the perspective of cosmology, since approximate shift symmetry on the inflaton φ, which is there is no need to worry about this sinflaton mode only broken by the holomorphic superpotential. This can be being too light (or tachyonic). This is the analogous accomplished by requiring that the Kähler potential has a feature that we saw with the S2 ¼ 0 nilpotency manifest inflaton shift symmetry and depends only on the condition; the sgoldstino S no longer contributes to real superfield B via (see also [7]) the bosonic evolution since it is constrained to be a 2 KðS; S¯; Φ; Φ¯ Þ¼SS¯ þ B : ð2:15Þ function of the fermions S ¼ðχsÞ2=ð2FsÞ. More- over, when we transition to local supersymmetry in Here we neglect terms linear in S and in B, since they may Sec. III, we can work in a unitary gauge for the be removed by a Kähler transform. gravitino where χs ¼ 0, in which case we will be The superpotential is a holomorphic function of the able to show that Φ ¼ φ is a pure real function. inflaton superfield Φ, therefore it cannot depend on the (2) The inflatino χϕ is no longer independent and is shift symmetric B (which contains the antiholomorphic Φ¯ ). rather proportional to the goldstino χs: Any dependence of the superpotential on Φ introduces deviation from the shift symmetry of the Kähler potential χ¯s 7 ϕ μ given in (2.15). The most general superpotential we can χ ¼ iσ ∂μΦ: ð2:11Þ F¯ s write is

This striking feature of the orthogonality/nilpotency WðS; ΦÞ¼fðΦÞS þ gðΦÞ: ð2:16Þ constraint might lead to a solution of the long- standing cosmological problem of gravitino-infla- Going beyond the assumption of a strict shift symmetry tino mixing which will be explained in detail in in (2.15), the most general form of the Kähler potential Secs. III and V. The importance of this is that in consistent with nilpotency and orthogonality is unitary gauge, both the inflatino χϕ and the goldstino s ¯ ¯ ¯ χ fermion vanish and the only relevant fermionic KðS; S; Φ; ΦÞ¼h0ðAÞSS þ h1ðAÞþh2ðAÞB degree of freedom is the massive gravitino. 2 þ h3ðAÞB ; (3) The auxiliary field is at least quadratic or higher in χs: 1 A ≡ ðΦ þ Φ¯ Þ: ð2:17Þ       2 s s s 2 ϕ χ¯ μ ν χ¯ 1 χ¯ 2 F ¼ −∂ν σ¯ σ ∂μΦ þ ∂ Φ: F¯ s F¯ s 2 F¯ s However, this expression can typically be simplified using field redefinitions and Kähler transformations; the details ð2:12Þ are given in Appendix A. Assuming the hiðAÞ functions are not pathological, (2.17) can be reduced to The fact that Fϕ vanishes in unitary gauge in a locally supersymmetric model leads to an interesting KðS; S¯; Φ; Φ¯ Þ¼SS¯ þ hðAÞB2: ð2:18Þ conclusion that the dependence of the off-shell ϕ bosonic potential on F is eliminated into the The function hðφÞ generates a nonminimal Kähler metric fermionic sector, as discussed in Sec. IV. for Φ and introduces an additional breaking of the shift By direct computation, one can show that the nilpotency condition (2.3) and orthogonality condition (2.9) imply a 7 third degree nilpotency condition [5] In [7], the superpotential was taken to be independent of the inflaton, leading to a flat inflaton potential. In that case, hφ_ i must 3 ¯ be inserted by hand, as expected since that EFT only aims to B ðx; θ; θÞ¼0: ð2:13Þ describe fluctuations about a fixed inflating background.

043516-4 COSMOLOGY WITH ORTHOGONAL NILPOTENT SUPERFIELDS PHYSICAL REVIEW D 93, 043516 (2016) symmetry. For simplicity, we set hðφÞ¼1 for our dis- A. Importance of unitary gauge cussion in Sec. V, though it might lead to interesting Local supersymmetry, as any gauge symmetry, requires a features for cosmological model building. choice of the gauge-fixing condition. The super-Higgs With regards to model building, it is sometimes more effect, where the gravitino ψ μ becomes massive by eating convenient to start with general models in the form a goldstino v, can be explained by the fact that the action with local supersymmetry requires a gravitino-goldstino ¯ ¯ ¯ ¯ KðS; S; Φ; ΦÞ¼h0ðAÞSS þ kðΦ; ΦÞ; mixing term, ¼ ðΦÞS þ ðΦÞ μ W f g ; ψ¯ γμv þ H:c:; ð3:1Þ S2 ¼ SB ¼ 0; ð2:19Þ where the goldstino v is some combination of spin-1=2 fermions. In the case of cosmological models with two ðΦ Φ¯ Þ where k ; might have a particular symmetry before superfields, a nilpotent stabilizer S and an unconstrained ðAÞ the constraints are imposed and the term h0 is present inflaton multiplet Φ, the goldstino field is in string-inspired models with warped geometry. For example, α-attractor models [16,17] have a hyperbolic 1 K=2 s ϕ ¯ v ¼ pffiffiffi e ðχ ∇ W þ χ ∇ϕWÞ; ð3:2Þ Kähler geometry encoded in kðΦ; ΦÞ, and one might 2 s therefore be interested in using these geometric variables directly instead of performing a Kähler transform to where ∇i is a Kähler-covariant derivative with respect to the simplify the Kähler potential. Examples with warped ith chiral multiplet. geometry are given in Sec. IV of [28], where Φ is a The possible unitary gauges for models with a nilpotent multiplet representing the volume of an extra dimension. multiplet and an unconstrained inflaton multiplet were Of course, (2.17) and (2.19) are physically equivalent, but discussed in [42]. It turns out that the standard unitary using variables with particular physics interpretations may gauge be preferable as a starting point. Indeed, using both variable choices might give extra insights into cosmo- v ¼ 0 ð3:3Þ logical model building. is not convenient for calculational purposes. The reason is that the uneaten linear combination of χs and χϕ has III. FROM GLOBAL TO LOCAL complicated couplings inherited from the high powers of SUPERSYMMETRY 2 χs interactions necessitated by the S ¼ 0 constraint. An In ordinary cosmological models with local supersym- alternative unitary gauge is9 metry and (at least) two chiral multiplets, the inflaton is an unconstrained chiral superfield and the inflatino is present χs ¼ 0; ð3:4Þ in the low energy spectrum.8 There is only one combi- nation of the spin-1=2 particles which can be removed by which has some calculational advantages since it removes s a gauge fixing condition, and the inflatino generically these higher order terms in χ . The gauge, however, leaves a remains mixed with gravitino. For this reason, the analysis mixing term between the gravitino and inflatino, of gravitino-inflatino production in models with two or 1 – μ K=2 ϕ more chiral multiplets is rather involved [35 37] (see ψ¯ γμ pffiffiffi e χ ∇ϕW: ð3:5Þ Sec. V below). As anticipated in (2.11), we now show how 2 the orthogonality condition SB ¼ 0 plus the appropriate ¼ 0 χs ¼ 0 gravitino gauge choice can remove these inflatino Note that one can set either v or in these complications. constructions, and one cannot realize both simultaneously ∇ ¼ 0 Compared to the previous section, we are now transi- unless ϕW . tioning from two-component Weyl notation to four- In this context, imposing the orthogonality constraint SB ¼ 0 component Majorana notation for the fermions in order is a particularly attractive option, since it suggests “ ” to make contact with the supergravity literature. the possibility of consistent inflatino-less cosmological models. Recall from (2.11) that in global supersymmetry, the vanishing of χs implies the vanishing of inflatino: 8There are inflationary models built from just one chiral multiplet [39–41], in which case the inflatino and the goldstino are identified. In such models, there is no inflatino in unitary 9Any gauge-fixing condition of local supersymmetry which gauge and therefore no inflatino-gravitino mixing. However, in depends only on chiral matter fermions and not on the gravitino is these models it is very difficult to achieve supersymmetry a unitary gauge, in the sense that there are no propagating ghost μ breaking at the end of inflation [41] and a second field typically degrees of freedom. This is as opposed to gauges like γ ψ μ ¼ 0 or μ has to be introduced. D ψ μ ¼ 0 which do necessitate propagating ghosts.

043516-5 FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016) χs ¼ 0 ⇒ χϕ ¼ 0: ð3:6Þ superconformal action with a complex vector Lagrange multiplier Ω, If this relation were also to hold in local supersymmetry, ½ΩSB þ ð Þ then we could find a unitary gauge choice where D H:c:; 3:11

v ¼ 0 and χs ¼ 0; ð3:7Þ the equations of motion for Ω impose the superfield constraint SB ¼ 0.10 That said, it is a bit more involved despite the fact that local gauge symmetry naively allows to show how the constraints arising from SB ¼ 0 are only one of these conditions. As we show below, (3.6) is modified in the local case, since this constraint equation indeed valid in local models, allowing us to achieve a very now involves terms containing the gravitino and the vector interesting class of inflatino-less cosmological models. auxiliary field of supergravity. Before proceeding on this, it is worth reflecting on why S2 ¼ 0 B. Imposing the constraints the nilpotent constraint gives expressions (2.3) and (3.10) which are equally valid in global and local super- In order to verify (3.6), we need to figure out how the symmetry. The local superconformal calculus and the rules SB ¼ 0 constraint and the corresponding equations (2.10), for multiplication of superfields are well known; for (2.11), and (2.12) are modified in models with local example, in the case of chiral multiplets, the comparison supersymmetry. between the global and local rules is described in chap- S2 ¼ 0 In the case of a single nilpotent constraint , the ter 16.2.1 of [43]. In general, any ordinary derivative that generalization from global to local supersymmetry is appears in the global case must be replaced by a super- known. At the level of the superconformal theory, the conformal derivative in the local case [defined, e.g., in constraint can be implemented via a Lagrange multiplier, a (16.34) and (16.37) of [43]]. These superconformal deriv- Λ chiral superfield [13]. By supplementing the super- atives introduce dependence on the gravitino and on the conformal action with vector auxiliary field which were absent in the global case, in principle modifying the meaning of S2 ¼ 0. Note, ½ΛS2 þ ð Þ F H:c:; 3:8 however, that all equations in (2.3) depend only on the undifferentiated spinor χs and undifferentiated scalar S. one can write down the superconformal action (3.8) in Therefore, there is no place in these algebraic equations to χs components, where the nilpotent chiral multiplet is (S, , replace ordinary derivatives with superconformal ones. s Λ χΛ Λ F ) and the Lagrange multiplier is ( , , F ). The This is a shortcut to reach the conclusion that the global corresponding locally superconformal action is given in constraints in (2.3) are valid in the local theory. Note that Λð Þ [23]. The field equation for x in the local superconfor- this same logic holds for any holomorphic constraint.11 mal theory follows and is given by In the case of SB ¼ 0, on the other hand, the constraints pffiffiffi 1 depend on supercovariant derivatives. This is expected 2 s − ðχsÞ2 þ 2ψ¯ γμ χs þ ψ¯ γμνψ 2 ¼ 0 since the global equations (2.10)–(2.12) depend on ordi- SF μ SPL 2 μPR νS : nary derivatives. One can see this explicitly in Appendix B ð3:9Þ where we derive the local version of SB ¼ 0 using the tensor calculus in supergravity [45,46]. The supercovariant By inspecting this equation together with similar equations derivatives of S and χs are for χΛ and FΛ, one finds that the solution of S2 ¼ 0 from (2.3) remains valid in the local theory, the most important i ˆ ¼ ∂ − ψ¯ χs ð Þ one being DμS μS 2 μ L; 3:12

2 s − ðχsÞ2 ¼ 0 ð Þ i SF : 3:10 ˆ χs ¼ χs − ð Þψ − sψ − χs ð Þ Dμ L Dμ L DS μR F μL 2 Aμ L; 3:13 We also see here that a nontrivial solution is possible only for Fs ≠ 0. This, in turn, means that the positive contri- where Dμψ is an ordinary covariant derivative of a spinor bution to the cosmological constant jFsj2 in a model with including the spin connection, and Aμ is the vector auxiliary local supersymmetry is a necessary consequence of the field. These additional terms present in the local case existence of a nontrivial nilpotent multiplet in a local theory. 10Alternatively, one can impose two real constraints, ðS þ ¯ ¯ The new orthogonality constraint SB ¼ 0 is no longer SÞB ¼ 0 and iðS − SÞB ¼ 0, using two real vector Lagrange holomorphic. This makes it more complicated to transition multiplier superfields. 11An interesting example of a holomorphic constraint is the from the global theory to the local one, since it requires a chiral orthogonality condition SΦ ¼ 0 [5,44]. This removes the D-term analysis in the superconformal theory. First, it is scalar modes from Φ but leaves the auxiliary field Fϕ,asis straightforward to prove that by supplementing the relevant for describing more general matter interactions [19].

043516-6 COSMOLOGY WITH ORTHOGONAL NILPOTENT SUPERFIELDS PHYSICAL REVIEW D 93, 043516 (2016) significantly complicate the derivation of the constrained Φ gauges, Fϕ is nonzero but SB ¼ 0 still implies that Fϕ does components. not have a bosonic part. This leads to an interesting property of the bosonic action in these inflatino-less models. C. Simplification in unitary gauge Crucially, however, there is a simplification to the A. A nonstandard potential SB ¼ 0 constraint equation if we work in unitary gauge In ordinary supergravity with unconstrained multiplets, where χs ¼ S ¼ 0. In that case one would expect the supergravity potential with two chiral multiplets S and Φ to take the form s ˆ χ ¼ 0 ⇒ DμS ¼ 0; ð3:14Þ ¯ ¯ ¯ 2 V ¼ eKð∇ Wgii∇¯W − 3jWj Þ;i¼ s; ϕ: ð4:1Þ ˆ χs ¼ − sψ ð Þ i i Dμ L F μ: 3:15 If the S multiplet is nilpotent, the potential is still given by At first glance, this might not seem like enough of a (4.1), simply evaluated at S ¼ 0. The reason this works is ˆ χs simplification, since Dμ L still involves the gravitino and that Fs is a full auxiliary field whose value is determined by s the (nonzero) auxiliary field F . The key point, though, is its equation of motion, just as in the unconstrained case. – ϕ that each fermionic term on the right-hand sides of (2.10) By contrast, if the Φ multiplet is orthogonal with F (2.12) contains as a factor at least one undifferentiated being fermionic, the potential takes the unusual form spinor χs, which goes to zero in unitary gauge. As shown χs K ss¯ ¯ ¯ 2 explicitly in Appendix B, this undifferentiated remains V ¼ e ð∇ Wg ∇¯W − 3jWj Þ; ∇ϕW ≠ 0: ð4:2Þ ˆ χs s s as a factor in the local expressions as well. So even if Dμ L is nonvanishing, any terms that depend on it do vanish in That is, the term ∇ϕW is not present in the potential, despite this unitary gauge. the fact that the bosonic part of ∇ϕW is nonvanishing. The We therefore conclude that when the local supersym- intuitive reason for this result is that Fϕ does not appear in χs ¼ 0 metry is gauge fixed with , the constrained values of the component action and has no equation of motion to χϕ ϕ the inflatino and auxiliary field F vanish: contribute a ∇ϕW term to the potential. χs ¼ 0 ⇒ χϕ ¼ Fϕ ¼ 0: ð3:16Þ B. Derivation from the off-shell action Moreover, the scalar component of Φ reduces to just a real Because (4.2) is sufficiently counterintuitive, it deserves inflaton: further explanation. One way to explain it is to start with off-shell supergravity in the form given in [25],12 namely Φ ¼ φ: ð3:17Þ −1L ¼ð i − i Þ ð ¯ i¯ − ¯ i¯ Þþ −1Lbook e off−shell F F gii¯ F F e ; The above logic is validated by the explicit computation G G ð Þ in Appendix B, which shows that nonzero terms 4:3 involving the gravitino, which are indeed possible in where principle, are in fact absent from the SB ¼ 0 constraint equations in unitary gauge. This now opens the pos- i ≡ − K=2 ii¯∇¯ ¯ þð i Þf ð Þ F e g i¯W F : 4:4 sibility of consistent inflatino-less cosmological models G G ð i Þf Lbook with local supersymmetry. In these expressions, FG depends on fermions and An interesting property of unitary gauge with vanishing is the full supergravity action in [43] with auxiliary fields inflatino and sinflaton is that the bosonic part of the vector eliminated on their equations of motion. For unconstrained auxiliary field multiplets, the first term in (4.3) vanishes on the Fi equations of motion. i ¯ bosonic ¼ ð ∂ i − ∂ ¯iÞðÞ Let us start with the case where S is nilpotent but Φ is Aμ Ki μz Ki¯ μz 3:18 2 unconstrained. If we are interested only in the bosonic action, we can use the expressions above together with the vanishes for Kähler potentials of the form (2.18), though constraint in (2.3) and deduce that the Fs and Fϕ equations not in the general case (2.19) prior to performing a Kähler of motion are still transform. 2 ¯ ¯ ¯ ϕ 2 ϕ¯ ¯ ¯ Fs ¼ −eK= gsi∇¯W; F ¼ −eK= g i∇¯W; ð4:5Þ IV. SUPERGRAVITY POTENTIAL WITH i i CONSTRAINED SUPERFIELDS except everywhere the condition In addition to the fact that the inflatino vanishes in unitary gauge, an interesting feature of (3.16) is that the auxiliary 12We replace the index α labeling chiral multiplets in [25,43] field Fϕ vanishes in this gauge as well. In more general by i here.

043516-7 FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016) S ¼ 0 ð4:6Þ V. INFLATINO-LESS MODELS WITH ORTHOGONAL NILPOTENT must be inserted. The bosonic potential in this case is MULTIPLETS simply As an example of the model building possibilities presented by inflatino-less constructions, consider super- ¼ Kð∇ ii¯∇¯ ¯ − 3j j2Þj ¼ ϕ ð Þ V e iWg i¯W W S¼0 i s; : 4:7 gravity models with a shift-symmetric Kähler potential as in (2.15) and a generic superpotential as in (2.16): Now consider the situation with an orthogonality con- Φ 1 straint on . It is convenient to separate the action into KðS; S¯; Φ; Φ¯ Þ¼SS¯ − ðΦ − Φ¯ Þ2; terms that depend on the auxiliary fields, 4 W ¼ fðΦÞS þ gðΦÞ; ¯ ¯ ¯ Lð ¯ Þ¼ð i − i Þ ð ¯ i − ¯ i Þ − i ¯ i ð Þ 2 F; F F FG gii¯ F FG FGgii¯FG; 4:8 S ¼ SB ¼ 0: ð5:1Þ and terms that are independent of Fi, In these models, violation of the inflaton shift symmetry enters only via the superpotential. i ¯ i¯ þ −1Lbook ð Þ FGgii¯FG e : 4:9 A. Action in unitary gauge Note that the contribution to the bosonic potential propor- In the unitary gauge with χs ¼ 0, the action correspond- 2 tional to j∇iWj is absent in (4.9). With the orthogonality ing to (5.1) is surprisingly simple: condition on Φ, the new and unusual situation is that the ϕ 1 off-shell auxiliary field F is constrained to be −1L ¼ ½ ðωð ÞÞ − ψ¯ γμνρ ψ þ L e 2κ2 R e μ Dν ρ SG;torsion ϕ ðφÞ 1 F ¼ 0; ð4:10Þ þ g ψ¯ γμνψ − ð∂φÞ2 þ 3κ2 2ðφÞ − 2ðφÞ bosonic 2 μ ν 2 g f : as implied by the local version of (2.12). Therefore Fϕ does ð5:2Þ not contribute in (4.8), The field-dependent gravitino mass is Lð ¯ Þj ⇒ s ¯ s¯ þ K=2ð s∇ þ ∇¯ ¯ ¯ s¯Þ F; F bosonic F gss¯F e F sW s¯WF ; ðφÞ¼ ðφÞκ2 ¼ ðφÞ −2 ð Þ m3=2 g g MPl ; 5:3 ð4:11Þ and we have taken the simple case where gðφÞ and fðφÞ are and the total bosonic potential comes from integrating out both real functions. Here, the gravitino kinetic term is Fs and from the terms in (4.9), leading to noncanonical in order to scale out the κ coupling, the spinor derivative Dμ is a vierbein-dependent spin connection, and K 2 2 L ; is the standard quartic in the gravitino term in V ¼ e ðj∇ Wj − 3jWj Þj ; ∇ϕW ≠ 0; ð4:12Þ SG torsion s S¼0 N ¼ 1 supergravity. There are numbers of ways to confirm this action. One way is to start from the results in [24,25], S2 ¼ 0 as claimed above and taking into account that . set χs ¼ 0 and b ¼ 0, and remove the ðg0Þ2 term from the The argument above can also be given using the off-shell potential as advocated in Sec. IV. An alternative approach is supergravity equations in the construction of [46]. In that to use the explicit action in [26] following from the general language, the part of the action relevant for the bosonic formula given in [25]. In particular the term proportional to potential is μνρσ ϵ ψ¯ μγνψ νAρ with the vector auxiliary field vanishes in models based on (5.1), as explained in (3.18). 1 ˆ ˆ ˆ ˆ ¯ 2 ˆ i ˆ j¯ − UU þðUW þ U WÞeK= þ K ¯h h The action in (5.2) describes the graviton, the gravitino 3 ij with a field-dependent mass term proportional to gðφÞ, ¯ þðˆ i þ ˆ i ¯ Þ K=2 ð Þ a canonically normalized inflaton φ, and an inflaton h DiW h Di¯W e : 4:13 potential: In the standard unconstrained situation, integrating out the VðϕÞ¼f2ðφÞ − 3κ2g2ðφÞ¼f2ðφÞ − 3M2 m2 ðφÞ: compensator field Uˆ and the auxiliary fields hˆ i leaves us Pl 3=2 with the supergravity potential in (4.1). If in this off-shell ð5:4Þ action we first take into account that hˆ ϕ is fermionic, though, then the contribution of the term ∇ϕW will drop out This potential has a de Sitter vacuum under the condition from the bosonic part of the action, leading to (4.12). that V>0 at V0 ≡∂V=∂φ ¼ 0, namely

043516-8 COSMOLOGY WITH ORTHOGONAL NILPOTENT SUPERFIELDS PHYSICAL REVIEW D 93, 043516 (2016) f2 − 3κ2g2 > 0 at ff0 − 3κ2gg0 ¼ 0: ð5:5Þ (6.1) below]. As a partial solution, recent inflationary models have been constructed such that ∇ϕW ¼ 0 at the There is no sinflaton, sgoldstino, or inflatino in the minimum of the potential [14–17]. In this case, there is no spectrum. mixing of the gravitino with the remaining spin-1=2 field as We can also look at general models in the form of (2.19) in (3.5) in χs ¼ 0 unitary gauge, at least at the minimum. where the geometry of the is important but This condition is still not quite satisfactory, though, since ¯ with canonical SS, using [24–26]. In such cases, there will ∇ϕW does not vanish exactly for oscillations about the be a few corrections to the action in (5.2): the kinetic term minimum of the potential, still leading to gravitino-inflatino of the real scalar will include dependence on kΦΦ¯ , the mixing. This complicates the whole analysis of creation of ðφÞ potential might have a nontrivial factor of eK in front, and matter at the end of inflation, since the equations of motion the spinor derivative of the gravitino might include a for the gravitino do not decouple from the leftover nonvanishing vector auxiliary field in (3.18). combination of the matter multiplets. For this reason, the fact that orthogonal nilpotent multiplets have no inflatino in B. Advantages for cosmology their spectrum is very promising. Supergravity actions based on (5.2) have various advan- tages for constructing inflationary models and for studying C. Gravitino dynamics reheating and matter creation after inflation. First, due to The super-Higgs mechanism in cosmology was studied the nilpotency and orthogonality constraints, there is only in [35], where the supersymmetry breaking scale was one real scalar inflaton, in agreement with current cosmo- shown to be equal to logical data consistent with a single scalar field driving inflation. Second, the three other scalars (complex sgold- α ¼ 3M2 ðH2 þ m2 Þ; ð5:8Þ stino and real sinflaton) which would be present for Pl 3=2 unconstrained multiplets are absent and therefore require no stabilization. Third, the absence of the inflatino sim- where H is the Hubble parameter. The equations of motion plifies the investigation of matter creation at the end of for the gravitino were derived in [48] for a model with one inflation. It is rather encouraging that these three desirable chiral inflaton multiplet, assuming that during inflation the features are present in a locally supersymmetric action. inflaton is real, and therefore terms depending on the vector To understand this importance of these features, it is auxiliary field Aμ drop from the bosonic evolution. worth reflecting on the complications present in previous Analogous equations for gravitino were given in [49]. models. In earlier nilpotent models with S2 ¼ 0 [14–19], Later, a more general form of the gravitino equations for the absence of the complex sgoldstino is well known. But models with any number of chiral multiplets was derived in because Φ was unconstrained, it was still necessary to [35], and the case of two chiral multiplets was treated in stabilize the sinflaton field b in order to ensure that the detail. As in Sec. III A, however, it was not possible to find evolution of the universe is driven by a single scalar a unitary gauge choice which would remove all mixing inflaton φ. For stabilizing the sinflaton, one can introduce terms between the gravitino and the chiral fermions. terms associated with the bisectional curvature of the Moreover, the vector auxiliary field played a significant Kähler manifold [47]. In particular, by adding terms of role in the gravitino equations of motion in a Friedmann- the form Robertson-Walker background, since one had to use ˆ i bosonic “hatted” time derivatives in the form ∂0 ¼ ∂0 − 2 γ5A0 . K ⊃ CðΦ; Φ¯ ÞSS¯ðΦ − Φ¯ Þ2; ð5:6Þ These complications are now absent for these new inflatino-less constructions. As emphasized above, we ϕ 2 ∼ ðΦ Φ¯ Þj sj2 can work in unitary gauge with χs ¼ χ ¼ 0 and the vector one can make the sinflaton heavy mb C ; F and proportional to the moduli space bisectional curvature auxiliary field is zero if we work with a Kähler potential in ∼ ðΦ Φ¯ Þ the form of (5.1), allowing us to avoid the assumptions RSS¯ΦΦ¯ C ; . Alternatively, terms in the Kähler poten- tial of the form made in [48]. As a result, the massive gravitino equations of motion are of the same kind as in [48,49], since despite K ⊃ ðΦ − Φ¯ Þ4 ð5:7Þ starting with two chiral multiplets S and Φ, the constraints imply a unitary gauge with just a massive gravitino and no have been shown to stabilize b [16]. In both cases, making other fermions. b heavy removes it from the cosmological evolution. In an expanding universe, the equations of motion for the Even if the sinflaton was constrained, however, the spin-1=2 and spin-3=2 components were derived in [35]. inflatino was still present in the dynamics, which intro- They were brought to a simpler form in [37], and in this duced significant complications in understanding the end of notation, the equations for the spin-1=2 longitudinal i inflation [35–37]. To our knowledge, there is no nilpotent component θ ¼ γ¯ ψ i and for the spin-3=2 transverse model in the literature with a heavy inflatino [see, however, component ψ~ T are

043516-9 FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016)

ˆ i 0 ˆ ~ discussion in Sec. IV, the standard supergravity scalar ½∂0 þ B þ iγ¯ kiγ¯ AθðkÞ¼0; ð5:9Þ   potential applies to this kind of inflatino-less construction. a_ γ¯0∂ þ γ¯i þ γ¯0 þ ψ Tð~Þ¼0 ð Þ ¯ 0 i ki 2 am3=2 ~ k ; 5:10 B. Relaxed constraints: S2 ¼ 0, SΦ is chiral An alternative mechanism for decoupling fermions from where we have assumed that m3=2 is real, a is the scale chiral multiplets was studied in [5], based on the constraints factor, γ¯μ are flat space gamma matrices, dots represent _ ≡ −1∂ ≡ 2 ¯ ¯ derivatives with respect to physical time f a 0f, H S ¼ 0; Dα_ ðSΦÞ¼0; ð6:2Þ a=a_ is the Hubble expansion rate, and such that SΦ¯ is a (composite) chiral multiplet. At the level _ 0 2 H þ γ¯ m_ 3 2 of global supersymmetry, it was shown that the first ˆ ¼ −1 − = A 2 2 ; component of the inflaton multiplet Φ ¼ φ þ ib is left 3 H þ m3 2 = unconstrained, but the expressions in (2.11) and (2.12) still 3 1 ˆ ˆ 0 ˆ hold, relating the inflatino and auxiliary field to Φ and the B ¼ − a_ A þ am3=2γ¯ ð1 þ 3AÞ: ð5:11Þ 2 2 goldstino χs. In particular in unitary gauge with χs ¼ 0, χϕ ϕ Φ i j and F both vanish. Because is a general complex field, Note that the extra fermionic terms ϒ ¼ gj ðχi∂0ϕ þ j these constraints do not lead to a nilpotency condition of χ ∂0ϕ Þ present in the gravitino equations with two or 3 i third degree (B ≠ 0), making it possible to write down more chiral multiplets in [35,37] are absent in our inflatino- more general Kähler potentials than in the SB ¼ 0 case. We less models since no such fermions appear in unitary gauge. verify the above features for local supersymmetry in χs ¼ 0 We see that the transverse modes ψ~ T have canonical gauge in Appendix C. kinetic terms, but the spatial part of the kinetic terms for If we impose a shift symmetry on the Kähler potential the longitudinal modes θ are modified by the matrix Aˆ . broken only via the superpotential, these models start with As recently emphasized in [7], these modifications to the “ ” – kinetic structure are known as the slow gravitino [50 54] KðS; S¯; Φ; Φ¯ Þ¼SS¯ þ kðΦ − Φ¯ ÞþcSS¯ðΦ − Φ¯ Þ2 þ; and are required to consistently embed supersymmetry in a (cosmological) fluid background. W ¼ fðΦÞS þ gðΦÞ; ð6:3Þ

where k is a function and c is a constant. Because Fϕ is VI. ALTERNATIVE INFLATINO-LESS constrained to be fermionic, one should again take out CONSTRUCTIONS ∇ϕW terms from the scalar potential as discussed in Given the above discussion, it is clearly desirable to have Sec. IV. Because the inflatino is no longer in the spectrum mechanisms to decouple the inflatino from the inflationary in unitary gauge, one need not impose ∇ϕW ¼ 0 at the dynamics. Here, we briefly present two alternative con- minimum to avoid inflatino-gravitino mixing. The sinfla- structions that do not rely on the SB ¼ 0 constraints, tion b is still in the spectrum, however, so one has to verify leaving a more complete study to future work. that it is properly stabilized. Models based on (6.3) are interesting for studies of the A. Linear S terms in the Kähler potential initial conditions for inflation. For example in [17],itwas found that the sinflaton evolved to the bottom of the valley S2 ¼ 0 In standard sgoldstino-less constructions with but at b ¼ 0 during cosmological evolution. Using the relaxed Φ unconstrained, the operator constraints in (6.2), such models would effectively achieve B3 ¼ 0 dynamically, and the absence of the inflatino would ⊃ ðS þ S¯ÞðΦ − Φ¯ Þ2 ð Þ K c 6:1 still simplify the analysis of matter creation during oscil- lations near the minimum of the potential. gives rise to an inflatino mass, where c is a constant. Note that this expression is linear in S, and the resulting inflatino VII. DISCUSSION mass is proportional to cFs. Since this term maintains the shift symmetry on φ, it does not introduce any additional In this paper, we used orthogonal nilpotent superfields to terms in the inflaton potential, though it does modify the consistently embed single-field inflation in nonlinearly dynamics of the sinflaton. To our knowledge, this term does realized local supersymmetry, thereby achieving infla- not appear in the literature on supergravity and inflation, tino-less models of early universe cosmology. Perhaps a though it is reminiscent of a Giudice-Masiero-like μ term more apt description is that we achieved inflatino-sinflaton- S2 ¼ [55]. It leads to a nondiagonal kinetic term KSΦ¯ which is and-sgoldstino-less cosmology, since the constraints present during the cosmological evolution at the stage when SB ¼ 0 not only decouple the inflatino, but project out the scalar bðxÞ still evolves. Note that (6.1) does not three of the four real scalars naively present in two chiral decouple Fϕ from the scalar potential, so unlike the multiplets S and Φ. This leaves one real scalar in the

043516-10 COSMOLOGY WITH ORTHOGONAL NILPOTENT SUPERFIELDS PHYSICAL REVIEW D 93, 043516 (2016) spectrum with an approximate shift symmetry, yielding a action has nonlinearly realized local supersymmetry. Based locally supersymmetric action compatible with cosmologi- on the results of this paper, it is now possible to construct cal data which favors single-field slow-roll inflationary interesting and elegant cosmological models using orthogo- scenarios. nal nilpotent superfields interacting with supergravity [38]. Part of the reason we have emphasized the inflatino-less We look forward to further investigations of the new aspect of these constructions is that the absence of a observational consequences that arise from inflatino-less sinflaton and sgoldstino is a feature already present in inflation. existing models.13 Even for unconstrained chiral multiplets in linearly realized supersymmetry, it is known that the ACKNOWLEDGMENTS sectional curvature RSSS¯ S¯ makes the sgoldstino heavy and the bisectional curvature RΦΦ¯ SS¯ makes the sinflaton heavy We are grateful to I. Antoniadis, E. Bergshoeff, J. J. [see Eqs. (12), (13), (31) of [47]]. There are also models Carrasco, C. Cheung, G. Dall’Agata, E. Dudas, D. based on vector multiplets in the Higgs phase [56], where Freedman, Y. Kahn, A. Kehagias, A. Linde, M. Porrati, half of a complex scalar is eaten by a massive gauge field, D. Roberts, A. Sagnotti, L. Senatore, F. Zwirner, A. Van decoupling the sinflaton and leaving only a real inflaton at Proyen and T. Wrase for discussions and collaborations in low energies. related projects. The work of S. F. is supported in part by To our knowledge, though, there is no previous super- INFN-CSN4-GSS. The work of R. K. is supported by the gravity construction where the inflatino does not participate SITP, and by the NSF Grant No. PHY-1316699. The work in cosmological dynamics. The fact that the inflatino is no of J. T. is supported by the U.S. Department of Energy longer an independent field—and is entirely absent in (DOE) under Cooperative Research Agreement No. DE- unitary gauge—is a real bonus for orthogonal nilpotent SC-00012567, by the DOE Early Career Research Program models, since it eliminates the inflatino-gravitino problem No. DE-SC-0006389, and by a Sloan Research Fellowship for studies of reheating after inflation [35–37]. Another from the Alfred P. Sloan Foundation. new feature in inflatino-less models using the orthogonal ¯ ¯ SB ¼ 0 or relaxed Dα_ ðSΦÞ¼0 constraints is the modified scalar potential where the contribution from ∇ΦW is APPENDIX A: GENERAL KÄHLER POTENTIAL absent. This unique feature turns out to be very useful WITHOUT A SHIFT SYMMETRY for building new inflatino-less cosmologies [38]. In this Appendix, we show how to reduce (2.17) to (2.18) Before concluding, we would like to add a comment here by using field redefinitions and Kähler transformations. on a consistency of this scenario with gravitino scattering 14 Repeating (2.17) for convenience, the most general Kähler unitarity. In the EFT language, the constraint equations potential consistent with S2 ¼ 0 and SB ¼ 0 is S2 ¼ SB ¼ 0 should be regarded as “infinitely relevant” operators, since they impose constraints at every dynamical ¯ ¯ ¯ 2 KðS; S; Φ; ΦÞ¼h0ðAÞSS þ h1ðAÞþh2ðAÞB þ h3ðAÞB ; scale. Because of this, one has to be careful when perform- 1 ing naive power counting, since the effective cutoff of these A ≡ ðΦ þ Φ¯ Þ: ð2:17Þ constructions is not the Planck scale, but rather the scale at 2 which scattering amplitudes violate unitarity, in particular By the orthogonality condition SB ¼ 0, though, we can longitudinal gravitino scattering. That said, the effective ¯ Λ4 ≃ ð 2 þ 2 Þ 2 freely covert Φ ↔ Φ inside of the function h0: cutoff of the theory is generically at H m3=2 MPl, [7,15,35], which in most cases is safely above any of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ ¯ ¯ ¯ scales considered during inflationary evolution. We reserve SSh0ðAÞ¼SSh0ðΦÞ¼SS h0ðΦÞh0ðΦÞ : ðA1Þ a detailed study of this point to future work, where we study under which circumstances the reheating epoch can be Because the holomorphic field redefinition described using constrained multiplets. We will also study how to couple these models to matter fields and whether S S → pffiffiffiffiffiffiffiffiffiffiffiffiffi ðA2Þ this might give new insights into the possible mechanisms h0ðΦÞ for reheating. It is tempting to notice that, taken at face value, these does not affect the nilpotent or orthogonality conditions, we models look almost too good to be true: a single inflaton as can use this freedom to set h0 ¼ 1. By invoking the square ordered by observations from the sky, yet the underlying root in (A2) we are implicitly assuming that h0ðφÞ is a positive function of φ over the relevant field space, otherwise S would have a wrong sign kinetic term. 13In addition, “inflatino-sinflaton-and-sgoldstino-less” does not exactly roll off the tongue. To further simplify the Kähler potential, we can perform 14Four-fermion interactions in models with nonlinearly real- a general Kähler transformation on (2.17) using a complex ized supersymmetry were studied in [44]. function j:

043516-11 FERRARA, KALLOSH, and THALER PHYSICAL REVIEW D 93, 043516 (2016) δ ¼ ðΦÞþ ðΦ¯ Þ ð Þ K j j : A3 i ϕ ¼ − ð ϕÞ − s þ χ¯ χs ð Þ H SRe F ibF 4 L L; B8 Expressing this in terms of Φ ¼ A þ iB and Φ¯ ¼ A − iB and Taylor expanding in B, we have 1 ¼ − ð ϕÞ − s þ χ¯ϕχs ð Þ K SIm F bF L L; B9 δK ¼ðjðAÞþjðAÞÞ þ iðj0ðΦÞ − j0ðAÞÞB 4 00 00 2 − ð ðAÞþ ðAÞÞB ð Þ i ϕ j j : A4 ¼ − ˆ φ − ˆ − χ¯ γ χs ð Þ vμ SDμ ibDμS 4 R μ L; B10 With j being a real function, we can set h1 ¼ 0. With (the derivative of) j being a pure imaginary function, we can set 1 ϕ 1 λ ¼ ðDSˆ þ FsÞχ − ðDˆ ðφ − ibÞþF¯ ϕÞχs ; ðB11Þ h2 ¼ 0. Thus, the only physical function is h3, which is a 2 R 2 L generic function of A and, in the context of cosmology, 1 1 must be chosen to satisfy slow-roll requirements. ¼ − ˆ ˆ μðφ − Þþ s ¯ ϕ þ χ¯ϕ ˆ χs þ χ¯s ˆ χϕ D iDμSD ib iF F RD L LD R: Rewriting h3 → h, the most general Kähler potential 4 4 consistent with nilpotency and orthogonality is (2.18), ðB12Þ repeated for convenience: The relevant supercovariant derivatives are KðS; S¯; Φ; Φ¯ Þ¼SS¯ þ hðAÞB2: ð2:18Þ i Thus, at the two-derivative level, any orthogonal nilpotent ˆ ¼ ∂ − ψ χs ð Þ DμS μS 2 μ L; B13 model for cosmology depends only on the superpotential ðϕÞ ðϕÞ functions f and g and the Kähler potential func- i ðϕÞ ˆ χs ¼ χs − ð Þψ − sψ − χs ð Þ tion h . Dμ L Dμ L DS μR F μL 2 Aμ L; B14


In this Appendix, we derive the local version of the where ψ μ is the gravitino and Aμ is the vector auxiliary field SB ¼ 0 constraints. From these, it is straightforward but of supergravity. Φ tedious to derive the components of the constrained The SB ¼ 0 constraint is now equivalent to superfield. Our main goal is to support the assertion in χϕ ϕ Sec. III C that the inflatino and auxiliary field F are C ¼ ζ ¼ H ¼ K ¼ vμ ¼ λ ¼ D ¼ 0: ðB16Þ both zero in unitary gauge χs ¼ 0. We start with multiplets written in the notation of [45]: Replacing supercovariant derivatives with ordinary ones and neglecting gravitino and supergravity auxiliary fields, S ¼ð − χs − s − s − ˆ 0 0Þ ð Þ S; i L; iF ; F ; iDμS; ; ; B1 these reduce to the global constraints [5,7], whose solution is (2.10)–(2.12). Solving these equations in the local case is Φ ¼ðφ þ − χϕ − ϕ − ϕ − ˆ ðφ þ Þ 0 0Þ ib; i L; iF ; F ; iDμ ib ; ; ; rather involved for a generic gravitino gauge choice. χs ¼ 0 ðB2Þ In unitary gauge for the gravitino, though, L and S ¼ 0 imply Φ¯ ¼ðφ − χϕ ¯ ϕ − ¯ ϕ ˆ ðφ − Þ 0 0Þ ð Þ ib; i R;iF ; F ;iDμ ib ; ; ; B3 ˆ ⇒ 0 ˆ χs ⇒ − sψ ˆ χs ⇒ − ¯ sψ DμS ; Dμ L F μL; Dμ R F μR; 1 ð Þ B ¼ ðΦ − Φ¯ Þ B17 2 i  1 as anticipated in (3.14). The components of SB simplify ¼ − χϕ − ð ϕÞ − ð ϕÞ − ˆ φ 0 0 ð Þ b; 2 ; Re F ; Im F ; Dμ ; ; ; B4 dramatically in this gauge:   1 ϕ SB ¼ð ζ λ Þ ð Þ SB ⇒ 0 0 − s − s 0 sχ − s ¯ ϕ ð Þ C; ;H;K;vμ; ;D : B5 ; ; ibF ; bF ; ; 2 F R; iF F : B18 The components of the composite (complex) vector multi- By assumption Fs ≠ 0, so setting the above expression to plet are zero implies ¼ ð Þ C Sb; B6 b ¼ 0; χϕ ¼ 0;Fϕ ¼ 0; ðB19Þ 1 ζ ¼ − Sχϕ − ibχs ; ðB7Þ in agreement with (3.16) and (3.17). The only nonzero 2 L component of Φ is α and we have


S ¼ð0 0 − s − s 0 0 0Þ ð Þ i ϕ ; ; iF ; F ; ; ; ; B20 ¼ ˆ Φ¯ − Φ¯ ˆ − χ¯ γ χs ð Þ vμ iSDμ i DμS 2 R μ L; C8 Φ ¼ðφ; 0; 0; 0; −i∂μφ; 0; 0Þ; ðB21Þ λ ¼ − ð ˆ þ sÞχϕ þ ð ˆ Φ¯ þ ¯ ϕÞχs ð Þ i DS F R i D F L; C9 ¯ Φ ¼ðφ; 0; 0; 0;i∂μφ; 0; 0Þ: ðB22Þ ˆ ˆ μ ¯ s ¯ ϕ i ϕ ˆ s i s ˆ ϕ D ¼ −2DμSD Φ þ 2F F − χ¯ Dχ − χ¯ Dχ : ðC10Þ Note that SΦ¯ ¼ð0; 0; −iφFs; −φFs; 0; 0; 0Þ is a chiral 2 R L 2 L R multiplet. For SΦ¯ to be chiral, we need APPENDIX C: RELAXED CONSTRAINTS IN ˆ P ζ ¼ 0;H¼ iK; −iDμC ¼ vμ; LOCAL SUPERSYMMETRY R λ ¼ D ¼ 0: ðC11Þ For completeness, we also consider the less restrictive ¯ constraint that SΦ is a chiral multiplet. The calculation χs ¼ 0 ¼ 0 Going to unitary gauge with L and S , we have the follows the same logic as above. The starting point is simplification S ¼ð − χs − s − s − ˆ 0 0Þ ð Þ  S; i L; iF ; F ; iDμS; ; ; C1 SΦ¯ ⇒ 0 0 − Φ¯ s −Φ¯ s 0 − sχϕ 2 s ¯ ϕ ; ; i F ; F ; ; iF R; F F Φ¯ ¼ðΦ¯ χϕ ¯ ϕ − ¯ ϕ ˆ Φ¯ 0 0Þ ð Þ ;i R;iF ; F ;iDμ ; ; ; C2  i ϕ þ sχ¯ γμψ ð Þ ¯ 2 F R μL : C12 SΦ ¼ðC; ζ;H;K;vμ; λ;DÞ: ðC3Þ

The components of SΦ¯ are Imposing the chirality constraints sets

ϕ ϕ C ¼ SΦ¯ ; ðC4Þ χ ¼ 0;F¼ 0; ðC13Þ

ϕ Φ ζ ¼ χ − Φ¯ χs ð Þ but leaves the complex unconstrained: iS R i L; C5 s s ϕ S ¼ð0; 0; −iF ; −F ; 0; 0; 0Þ; ð Þ H ¼ iðSF¯ − Φ¯ FsÞ; ðC6Þ C14 Φ ¼ðΦ 0 0 0 − ∂ Φ 0 0Þ ð Þ K ¼ −ðSF¯ ϕ þ Φ¯ FsÞ; ðC7Þ ; ; ; ; i μ ; ; : C15

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