Generalized Fayet–Iliopoulos D-terms in supergravity and inflation driven by supersymmetry breaking

1. Introduction

In this paper, we present in a self-contained way the formalism that allows to generalize D-terms in supergravity with the same bosonic component given by a Fayet–Iliopoulos (FI) term, proportional to a D-auxiliary field of an abelian vector multiplet with a general scalar field dependent coefficient.^1,2,3,4 We then discuss their physical applications in constructing models based on, or involving, D-term inflation ‘by supersymmetry breaking’, where the inflaton is identified with the partner of the goldstone fermion of spontaneous supersymmetry breaking, the goldstino, while its pseudoscalar partner is absorbed into the associated gauge field that becomes massive.² The latter gauges an R-symmetry in the sense that the gravitino is charged. A short version of this review was sent in the IJMPA Special Issue: Beyond Classical Gravity and Field Theory— Contributions from the Balkan Congress ‘Fair’.

The above framework of the so-called inflation by supersymmetry breaking allows to address in a natural way the generic problems that inflationary models in supergravity suffer, such as fine-tuning to satisfy the slow-roll conditions, large field initial conditions that break the validity of the effective field theory, and stabilization of the (pseudo) scalar companion of the inflaton arising from the fact that bosonic components of superfields always come in pairs. A solution to these problems was proposed in the recent years by identifying the inflaton with the scalar partner of the goldstino offering a direct connection between supersymmetry breaking and inflation, in the presence of a gauged R-symmetry.⁵ Indeed, in the case of F-term driven supersymmetry breaking, the superpotential is linear and the slow roll conditions are automatically satisfied. Moreover, inflation arises at a plateau around the maximum of the scalar potential (hill-top), so that no large field initial conditions are needed, while the pseudo-scalar companion of the inflaton is absorbed into the R-gauge field that becomes massive, leading the inflaton as a single scalar mode present in the low-energy spectrum. This model allows also the presence of a realistic minimum describing our present Universe with an infinitesimal positive vacuum energy arising due to a cancellation between F- and D-term contributions to the scalar potential, along the lines of Refs. 6 and 7. Finally, it predicts a rather small ratio of tensor-to-scalar primordial fluctuations $r\lesssim 10^{-4}$.

This paper is organized as follows. In Sec. 2, we review conformal supergravity in superspace.⁸ This can be regarded as a superspace version of superconformal tensor calculus. In Sec. 3, we review FI terms in supergravity. Section 4 introduce applications of supergravity actions with FI terms to inflation model buiding. Section 5 gives conclusion and outlook.

Throughout this review, we use notations and conventions of Wess–Bagger,⁹ unless otherwise mentioned.

2. Conformal Supergravity in Superspace

2.1. Formalism

Supergravity is normally required to be invariant under super-diffeomorphism on curved superspace and local super-Poincaré transformations on each tangent superspace. Conformal supergravity uses the superconformal transformations instead of the super-Poincaré ones. Super-Poincaré invariant supergravity theories are obtained by gauge-fixing.

We first briefly review superconformal transformations. They consist of super-translations, Lorentz transformations, dilatations, chiral $U(1)$ rotations, and conformal super-translations and all of them are defined in a flat superspace. Translations shift the origin of a flat superspace, while the other four not. We then call the latter four the origin-preserving transformations. In terms of curved superspace, they act on the tangent superspace of each point and hence they must be gauged (local).

Let us denote the generators of superconformal transformations collectively by $\{G_{\underline{\mathcal{𝒜}}}\}$, which consist of

where we introduced collective index A that covers vector, undotted and dotted indices: $P_A=(P_a,Q_{\alpha },\overline{Q}{}^{\dot{\alpha }})$, $K_A=(K_a,S_{\alpha },\overline{S}{}^{\dot{\alpha }})$. The underbarred calligraphic index $\underline{\mathcal{𝒜}}$ in the generator $G_{\underline{\mathcal{𝒜}}}$ runs over all possible generators listed above. The generators are closed under commutation relations to form the superconformal algebra $[G_{\underline{\mathcal{𝒜}}},G_{\underline{\mathcal{ℬ}}}]=-f_{\underline{\mathcal{𝒜}}\underline{\mathcal{ℬ}}}{}^{\underline{\mathcal{𝒞}}}{G}_{\underline{\mathcal{𝒞}}}$, where the structure constants are given concretely in (A.1).

If we want to consider actions invariant under gauged internal transformations, we also need to add their generators $\{T_I\}$ to the list of the generators above. In accord, we let the collective indices $\underline{\mathcal{𝒜}},\underline{\mathcal{ℬ}},\dots$ cover $\{T_I\}$. Here, ‘internal’ means that $\{T_I\}$ commute with the superconformal generators and satisfy $[T_I,T_J]=f_{IJ}{}^K{T}_K$.

Transformation properties of supergravity actions are described with variations (infinitesimal transformations) of superfields under the gauge transformations. We denote variations by^a

where the unbarred calligraphic index like $\mathcal{𝒜}$ means that it does not refer to the translation $\{P_A\}$. The symbol ${\delta }_{♠}(♣)$ denotes the variation under the generator $♠$ with parameters $♣$ that are functions on curved superspace. The variations are defined in such a manner that they are closed under commutation relations :

Since the variations under the superconformal transformations are defined in flat superspace, we need to proceed to the next step of deforming them into variations defined in curved superspace. The idea is to replace ${\delta }_P$ for gauged translation by Lie derivatives ${\delta }_L$ (infinitesimal diffeomorphisms) on curved superspace. However, the Lie derivative involves derivatives of parameters, and is inconvenient for construction of invariant actions.

We then further deform the Lie derivative ${\delta }_L$ into the infinitesimal parallel transport ${\delta }_{\parallel }$. This must be so defined that it does not involve derivatives of parameters. For this, we introduce connection superfields $H_M{}^{\underline{\mathcal{𝒜}}}$ for the superconformal generators. More explicitly, the connection superfield for each generator is denoted by

In particular, $E_M{}^A$ is called the vierbein. This will be used to relate indices for curved superspace and those for tangent superspace.

Now, the definition of ${\delta }_{\parallel }$ is

The point is that the connections also transform under $G_{\mathcal{𝒜}}$

The new variation

is closed under commutations relations, and satisfies the same commutation relations as the superconformal algebra except for those involving only parallel transports,

$$[\underline{\delta }({ϵ}_1^{\underline{\mathcal{𝒜}}}),\delta ({ϵ}_2^{\mathcal{ℬ}})]=\underline{\delta }(-{ϵ}_2^{\mathcal{ℬ}}{ϵ}_1^{\underline{\mathcal{𝒜}}}{f}_{\underline{\mathcal{𝒜}}\mathcal{ℬ}}{}^{\underline{\mathcal{𝒞}}}),$$(7)

$$[{\delta }_{\parallel }({ϵ}_1^A),{\delta }_{\parallel }({ϵ}_2^B)]=\underline{\delta }(-{ϵ}_2^B{ϵ}_1^A{R}_{AB}{}^{\underline{\mathcal{𝒞}}}),$$(8)

where the new structure “constants” $R_{AB}{}^{\underline{\mathcal{𝒞}}}$ are the curvature superfields:

$$R_{AB}{}^{\underline{\mathcal{𝒞}}}=E_A{}^M{E}_B{}^N{R}_{MN}{}^{\underline{\mathcal{𝒞}}},$$(9)

$$R_{MN}{}^{\underline{\mathcal{𝒞}}}={\partial }_M{H}_N{}^{\underline{\mathcal{𝒞}}}-{\partial }_N{H}_M{}^{\underline{\mathcal{𝒞}}},$$(10)

$$-(E_N{}^B{H}_M{}^{\mathcal{𝒜}}-E_M{}^B{H}_N{}^{\mathcal{𝒜}})f_{\mathcal{𝒜}B}{}^{\underline{\mathcal{𝒞}}}-H_N{}^{\mathcal{ℬ}}{H}_M{}^{\mathcal{𝒜}}{f}_{\mathcal{𝒜}\mathcal{ℬ}}{}^{\underline{\mathcal{𝒞}}}.$$(11)

Covariant superfields and covariant derivatives

We introduce an important class of superfields, called covariant superfields. This is defined by the property that $\delta ({ϵ}^{\mathcal{𝒜}})\mathrm{\Phi }$ does not contain derivatives of ${ϵ}^{\mathcal{𝒜}}$. It therefore makes sense to denote $\delta ({ϵ}^{\mathcal{𝒜}})\mathrm{\Phi }={ϵ}^{\mathcal{𝒜}}{G}_{\mathcal{𝒜}}\mathrm{\Phi }$, which defines the field $G_{\mathcal{𝒜}}\mathrm{\Phi }$ by separating the parameters.

The covariance is defined in terms of $\delta$, namely in terms of the variation under the origin-preserving superconformal transformations and the internal gauge transformations. Since they do not shift the origin in the tangent superspace, it is natural to assume that each generator $G_{\mathcal{𝒜}}\mathrm{\Phi }$ is represented as a matrix multiplication on $\mathrm{\Phi }$.

We give several examples of $G_{\mathcal{𝒜}}\mathrm{\Phi }$. For Lorentz generators, typical irreducible representations of the Lorentz algebra are given by

Let $\mathrm{\Phi }$ be in an irreducible representation of the Lorentz algebra. Since dilation and chiral $U(1)$ rotation $D,A$ commute with the Lorentz generators, Schur’s lemma implies that $D\mathrm{\Phi },A\mathrm{\Phi }$ become just multiplication by numbers. We then introduce the weights $(\mathrm{\Delta },w)$ of $\mathrm{\Phi }$ by : We then introduce a class of covariant superfields, called primary superfields. This is defined by

Let us consider a covariant superfield $\mathrm{\Phi }$ without any indices of curved superspace such as $M,N,\dots$. Then, the variation ${\delta }_{\parallel }({ϵ}^A)\mathrm{\Phi }$ under parallel transport reads

where we introduced the covariant derivative, Note that ${\nabla }_M$ is well-defined for any superfield as long as $\delta ({\xi }^{\mathcal{𝒜}})$ is defined.

To obtain the covariant variation (17) under parallel transport, the assumption that $\mathrm{\Phi }$ has no curved superspace indices was crucial. In what follows, we make this assumption for any covariant superfield.

In terms of the covariant derivatives, we introduce chiral and anti-chiral superfields by

Curvature constraints

The curvature superfields have too many degrees of freedom, so we normally impose constraints on them. Butter⁸ adopted the following curvature constraints:

where we introduced the explicit symbol for each generator:

$$R_{AB}{}^{\underline{\mathcal{𝒞}}}:R_{AB}{(P)}^C,R_{AB}{(M)}^{ba},R_{AB}(D),R_{AB}(A),R_{AB}{(K)}^C,{\mathcal{ℛ}}_{AB}^I.$$(24)

These constraints have a similar structure as those for the super Yang–Mills field strength with global supersymmetry. We can then solve the Bianchi identities under the constraints in a similar manner as in globally supersymmetric Yang–Mills case. Indeed, we can introduce “gaugino” superfields through Here $W_{\underline{\alpha }}$ are the generator-valued superfields, which can be expressed in terms of totally symmetric multi-spinor superfields $W_{\alpha \beta \gamma }$ and $\overline{W}{}_{\dot{\alpha }\dot{\beta }\dot{\gamma }}$ and gaugino superfields ${\mathcal{𝒲}}^{\alpha I},{\overline{\mathcal{𝒲}}}_{\dot{\alpha }}{}^I$ associated with the internal generators $\{T_I\}$,

$$W^{\alpha }=-W^{\alpha \beta \gamma }{M}_{\gamma \beta }-\frac{1}{2}{\nabla }^{\dot{\alpha }\gamma }{W}_{\gamma }{}^{\alpha \beta }{K}_{\beta \dot{\alpha }}+\frac{1}{2}{\nabla }^{\gamma }{W}_{\gamma }{}^{\alpha \beta }{S}_{\beta }-{\mathcal{𝒲}}^{\alpha I}{T}_I,$$(26)

$${\overline{W}}_{\dot{\alpha }}={\overline{W}}_{\dot{\alpha }\dot{\beta }\dot{\gamma }}{M}^{\dot{\gamma }\dot{\beta }}+\frac{1}{2}{\nabla }_{\alpha \dot{\gamma }}\overline{W}{}^{\dot{\gamma }}{}_{\dot{\alpha }\dot{\beta }}{K}^{\dot{\beta }\alpha }-\frac{1}{2}{\bar{\nabla }}_{\dot{\gamma }}\overline{W}{}^{\dot{\gamma }}{}_{\dot{\alpha }\dot{\beta }}{\bar{S}}^{\dot{\beta }}-{\overline{\mathcal{𝒲}}}_{\dot{\alpha }}{}^I{T}_I.$$(27)

Just as the super Yang–Mills gaugino superfields, $W_{\alpha \beta \gamma }$ and $\overline{W}{}_{\dot{\alpha }\dot{\beta }\dot{\gamma }}$ are chiral primary of weights $(3∕2,1)$ and anti-chiral primary of weights $(3∕2,-1)$, respectively. Furthermore, just as in globally supersymmetric super Yang–Mills case, the generator-valued superfields $W_{\underline{\alpha }}$ satisfy the reality condition in the following form : Note that the covariant derivatives act on the coefficient superfields in $W$. The anti-commutation relations in (28) are just composed of vector conformal translations $K_a$ and the internal generators $T_I$.

The vector–vector curvatures are expressed by

where we introduced the generator-valued curvature by Expanding this, we can express all vector-vector curvatures in covariant derivatives of $W_{\alpha \beta \gamma },\overline{W}{}_{\dot{\alpha }\dot{\beta }\dot{\gamma }}$. It provides various constraints on the vector-vector curvatures. For example, the vector torsion must vanish $R_{ab}{(P)}^c=0$.

Invariant integrals

Invariant actions are constructed with the invariant integrals. Let $(x^m,{\theta }^{\mu },{\bar{\theta }}_{\dot{\mu }})$ be a coordinate set of a curved superspace. Let us define the densities,

They transform as covariant primary superfields:

$$M_{ab}E=K_{A}E=0,DE=-2E,AE=0,$$(32)

$$M_{ab}\mathcal{ℰ}=K_A\mathcal{ℰ}=0,D\mathcal{ℰ}=-3\mathcal{ℰ},A\mathcal{ℰ}=-2i\mathcal{ℰ},$$(33)

$$M_{ab}\overline{\mathcal{ℰ}}=K_A\overline{\mathcal{ℰ}}=0,D\overline{\mathcal{ℰ}}=-3\overline{\mathcal{ℰ}},A\overline{\mathcal{ℰ}}=2i\overline{\mathcal{ℰ}}.$$(34)

Let $𝕍$ be a real scalar primary superfield of weights $(\mathrm{\Delta },w)=(2,0)$, $𝕎$ be a scalar chiral primary superfield of weights $(3,2)$, and $\overline{𝕎}$ be a scalar antichiral primary superfield of weights $(3,-2)$. We also assume that $𝕍,𝕎,\overline{𝕎}$ are all neutral (invariant) under $\{T_I\}$. Then, the following integrals are invariant under both ${\delta }_{\parallel }$ and $\delta$ :

The first integral is called the D-type integral, and the second and third are called the F-type integrals. They are related by They can be rewritten in terms of the lowest component projection $\theta =\bar{\theta }=0$ of spinorial covariant derivatives of $𝕍,𝕎$. The F-type integrals read

$$I_F[𝕎]=-\frac{1}{4}\int d^{4}xe({\nabla }^2𝕎|-2i{\bar{\psi }}_a\overline{\sigma }{}^a\nabla 𝕎|+4{\bar{\psi }}_a\overline{\sigma }{}^{ab}{\bar{\psi }}_b𝕎|),$$(37)

$$I_{\bar{F}}[\overline{𝕎}]=-\frac{1}{4}\int d^{4}xe({\overline{\nabla }}^2\overline{𝕎}|-2i{\psi }_a\sigma {}^a\overline{\nabla 𝕎}|+4{\psi }_a\sigma {}^{ab}{\psi }_b\overline{𝕎}|)$$(38)

and the D-type integral reads where the symbol “$|$” means setting $\theta =\bar{\theta }=0$ as usual, which we call the lowest component projection. Therefore, the invariant integrals are functionals of space–time fields.

Here we explain some symbols in (39). We defined the vierbein $e_m{}^a$ and the gravitino ${\psi }_m{}^{\underline{\alpha }}$ by

and ${\psi }_a{}^{\underline{\alpha }}=e_a{}^m{\psi }_m{}^{\underline{\alpha }}$. The density e is defined as $e=det{e}_m{}^a$. The scalar curvature $R(\omega )$ and the spinor torsions $T_{bc}{}^{\underline{\alpha }}$ are defined by

$$R(\omega )=e_b{}^m{e}_a{}^n({\partial }_m{\omega }_n{}^{ba}-{\partial }_n{\omega }_m{}^{ba}-{\omega }_m{}^{bc}{\omega }_{nc}{}^a+{\omega }_n{}^{bc}{\omega }_{mc}{}^a),$$(41)

$$T_{ab}{}^{\alpha }=\frac{1}{2}{e}_a{}^m{e}_b{}^n{\partial }_m{\psi }_n{}^{\alpha }+\frac{1}{4}{\omega }_a{}^{dc}{({\psi }_b{\sigma }_{cd})}^{\alpha }+\frac{1}{4}{b}_a{\psi }_b{}^{\alpha }-\frac{1}{2}i{c}_a{\psi }_b{}^{\alpha }-(a\leftrightarrow b),$$(42)

$${\bar{T}}_{ab\dot{\alpha }}=\frac{1}{2}{e}_a{}^m{e}_b{}^n{\partial }_m{\bar{\psi }}_{n\dot{\alpha }}+\frac{1}{4}{\omega }_a{}^{dc}{({\bar{\psi }}_b{\bar{\sigma }}_{cd})}_{\dot{\alpha }}+\frac{1}{4}{b}_a{\bar{\psi }}_{b\dot{\alpha }}+\frac{1}{2}i{c}_a{\bar{\psi }}_{b\dot{\alpha }}-(a\leftrightarrow b),$$(43)

where the spin connection ${\omega }_m{}^{ab}$, which is defined as ${\omega }_m{}^{ab}:={\mathrm{\Omega }}_m{}^{ab}|$, is not independent from the other connection fields, but given by where each term is defined by Here we introduced $b_a=e_a{}^m{B}_m|$. This is a consequence of the curvature constraint $R_{ab}{(P)}^c=0$.

Note that the first line of (39) is the sum of the kinetic terms of the graviton and gravitino, multiplied by an overall factor $𝕍|$.

SUSY transformation

Let us focus on parallel transports with spinor parameters, which we denote by

where $\underline{\alpha }$ runs over $\alpha ,\dot{\alpha }$. The invariant integrals are of course invariant under ${\delta }_{ϵ}$. In particular, the lowest component projections of the spinor covariant derivatives in the formulae (37)–(39) transform under ${\delta }_{ϵ}$ as where $X$ collectively denotes the spinor covariant derivatives of $𝕍$, $𝕎$ appearing in (37)–(39). We call the lowest component projection ${\delta }_{ϵ}X|$ of the transformed superfield ${\delta }_{ϵ}X$ the SUSY transformation of the projection $X|$.

We are free to extend this definition of the SUSY transformation to the lowest component projection of any superfield (for example the connection superfields), as long as ${\delta }_{ϵ}$ is defined for the superfield. This extension yields the SUSY transformation of the vielbein, gravitino and vector gauge fields in (37)–(39). It is clear that the SUSY transformation law (48) holds for any covariant superfield $X$.

Functions of superfields

The spinor covariant derivatives in the formulae (37)–(39) are listed as

where f denotes $𝕎,𝕍$ in the F-type, D-type integrals, respectively. In applications, f is a function of primary superfields, which we denote by $\{{\mathrm{\Phi }}^i\}$. They may be charged under gauged internal generators $\{T_I\}$ in general. Applying the Leibniz rule repeatedly, we can show that the spinor covariant derivatives in the list (49) can be written in the following fields : Explicit expansion formulae of spinor covariant derivatives of f are given in App. A. Note that the last two lines in the list contain the gaugino superfields $W_{\underline{\alpha }}$. Since ${\mathrm{\Phi }}^i$ is primary, we have $W_{\underline{\alpha }}{\mathrm{\Phi }}^i={\mathcal{𝒲}}_{\underline{\alpha }}^I{T}_I{\mathrm{\Phi }}^i$, which explains the absence of $W_{\underline{\alpha }}$ in the list (49) for the neutral $𝕍,𝕎$.

Supermultiplets

An important property of the list (50) is that the fields in the list are closed under SUSY transformations, which means that SUSY transformations of each field in the list (50) can be written as a linear combination of the fields in (50). We therefore call the list (50) the supermultiplet associated with the superfield  ${\mathrm{\Phi }}_i$, and the members of the list the component fields. In general, a supermultiplet is a list of fields which are closed under SUSY transformations. The one-to-one correspondence between the multiplet (50) and the multiplet in superconformal tensor calculus was demonstrated in Ref. 10 and 11.

Chiral superfields and chiral projection

A chiral primary superfield $\mathrm{\Phi }$ of weights $(\mathrm{\Delta },w)$ is defined by

Combining this with the primarity condition $K_A\mathrm{\Phi }=0$ yields $\{K_A,{\overline{\nabla }}_{\dot{\alpha }}\}\mathrm{\Phi }=0$. Nontrivial among them is actually $\{{\overline{S}}_{\dot{\alpha }},{\overline{\nabla }}_{\dot{\beta }}\}\mathrm{\Phi }=0$, which gives the constraint The supermultiplet of the chiral superfield $\mathrm{\Phi }$, the chiral multiplet, reads where ${\overline{\mathcal{𝒲}}}_{\alpha }{}^I|,\overline{\nabla \mathcal{𝒲}}{}^I|$ are the gauginos, the D-auxiliary fields, respectively, as will be explained in more detail later. One can see that the chiral multiplet is specified by giving $(\mathrm{\Phi }|,{\nabla }_{\alpha }\mathrm{\Phi }|,{\nabla }^2\mathrm{\Phi }|)$.

Similarly, the constraint on the weights $(\mathrm{\Delta },w)$ of an anti-chiral superfield $\mathrm{\Phi }$ reads

Its multiplet structure reads The last component field is zero because it is defined by ${\nabla }^{\alpha }\overline{\nabla }{}^2{\nabla }_{\alpha }-8W\nabla$. One can see again that the anti-chiral multiplet is specified by giving $(\mathrm{\Phi }|,{\overline{\nabla }}_{\dot{\alpha }}\mathrm{\Phi }|,\overline{\nabla }{}^2\mathrm{\Phi }|)$.

The chiral projection T on a primary superfield $\mathrm{\Phi }$ is defined by

Obviously, it is named after the property that $T(\mathrm{\Phi })$ is chiral, though T is not a ‘projector’ since it is not idempotent, $T^2\ne T$.

The requirement that $T(\mathrm{\Phi })$ be primary for primary $\mathrm{\Phi }$ yields :

This comes from $0=[\overline{S}{}^{\dot{\beta }},\overline{\nabla }{}^2]\mathrm{\Phi }$. Note that the other conditions are trivially satisfied.

The component expansion of the chiral projection of $\mathrm{\Phi }$ can be expressed in the component fields of $\mathrm{\Phi }$ as follows :

where we assumed just for simplicity that $\mathrm{\Phi }$ is neutral under $\{T_I\}$. In particular, if $\mathrm{\Phi }$ is anti-chiral, they become

In a similar manner, the anti-chiral projection $\overline{T}$ is defined by

The requirement that $\overline{T}(\mathrm{\Phi })$ be primary for primary $\mathrm{\Phi }$ yields The component expansion of the anti-chiral projections of $\mathrm{\Phi }$ can be expressed in the component fields of $\mathrm{\Phi }$ as follows : where we assumed just for simplicity that $\mathrm{\Phi }$ is neutral under $\{T_I\}$. In particular, if $\mathrm{\Phi }$ is chiral, they become

Gauge multiplets

Let us consider the superfields ${\mathcal{𝒲}}_{\alpha }^I,\overline{\mathcal{𝒲}}{}^{\dot{\alpha }I}$, which have already introduced as the $T_I$-components of $W_{\underline{\alpha }}$ as a consequence of the curvature constraints. ${\mathcal{𝒲}}_{\alpha }^I$ is chiral primary of weights $(3∕2,1)$ and $\overline{\mathcal{𝒲}}{}^{\dot{\alpha }I}$ is anti-chiral primary of weights $(3∕2,-1)$. By their chirality, the nontrivial components are

Let us consider the second component of ${\mathcal{𝒲}}^I$. This can be decomposed into the symmetric and anti-symmetric parts with respect to $\alpha \beta$. The symmetric part is nothing but the curvature $2{\mathcal{ℛ}}_{\alpha \beta }^I={\nabla }_{\beta }{\mathcal{𝒲}}_{\alpha }^I+{\nabla }_{\alpha }{\mathcal{𝒲}}_{\beta }^I$, while the anti-symmetric part ${\epsilon }_{\alpha \beta }\nabla {\mathcal{𝒲}}^I$ is new. We denote this by $D^I$. A similar argument holds for ${\overline{\mathcal{𝒲}}}^I$. Therefore, the first two component fields can be expressed as

$${\mathcal{𝒲}}_{\alpha }^I|=:{\lambda }_{\alpha }^I,{\nabla }_{\beta }{\mathcal{𝒲}}_{\alpha }^I|={\widehat{F}}_{\beta \alpha }^I+i{\epsilon }_{\beta \alpha }{D}^I,$$(66)

$$\overline{\mathcal{𝒲}}{}^{\dot{\alpha }I}|=:\overline{\lambda }{}^{\dot{\alpha }I},\overline{\nabla }{}^{\dot{\beta }}\overline{\mathcal{𝒲}}{}^{\dot{\alpha }I}|=-\widehat{F}{}^{\dot{\beta }\dot{\alpha }I}+i{\epsilon }^{\dot{\beta }\dot{\alpha }}{D}^I,$$(67)

where we introduced

$${\widehat{F}}_{ab}^I:={\mathcal{ℛ}}_{ab}^I|,{\widehat{F}}_{\alpha \beta }={({\sigma }^{ba}\epsilon )}_{\alpha \beta }{\widehat{F}}_{ab}^I,{\widehat{F}}^{\dot{\alpha }\dot{\beta }}={(\overline{\sigma }{}^{ba}\epsilon )}^{\dot{\alpha }\dot{\beta }}{\widehat{F}}_{ab}^I,$$(68)

$$D^I:=\frac{1}{2}\nabla {\mathcal{𝒲}}^I|=\frac{1}{2}\overline{\nabla \mathcal{𝒲}}{}^I,$$(69)

where the second identity in the definition of $D^I$ is the reality condition $\nabla {\mathcal{𝒲}}^I=\overline{\nabla \mathcal{𝒲}}{}^I$ from (28). According to the definition of the curvature superfields, ${\widehat{F}}_{ab}$ is given by:

$${\widehat{F}}_{ab}^I=e_a{}^m{e}_b{}^n{F}_{mn}^I+\frac{1}{2}i{\psi }_b{\sigma }_a\overline{\lambda }{}^I-\frac{1}{2}i{\overline{\psi }}_b{\overline{\sigma }}_a\lambda {}^I-\frac{1}{2}i{\psi }_a{\sigma }_b\overline{\lambda }{}^I+\frac{1}{2}i{\overline{\psi }}_a{\overline{\sigma }}_b\lambda {}^I,$$(70)

$$F_{mn}^I:={\partial }_m{\mathcal{𝒜}}_n-{\partial }_n{\mathcal{𝒜}}_m+A_n^K{\mathcal{𝒜}}_m^J{[T_J,T_K]}^I.$$(71)

Namely, it is nothing but the field strength augmented by fermions.

The remaining components ${\nabla }^2{\mathcal{𝒲}}^I|,\overline{\nabla }{}^2\overline{\mathcal{𝒲}}{}^I|$ are determined by the gauginos,

This is also a consequence of the reality condition $\nabla {\mathcal{𝒲}}^I=\overline{\nabla \mathcal{𝒲}}{}^I$.

Later on, we will also use the superfield ${\mathcal{𝒱}}_D:=\nabla {\mathcal{𝒲}}^I$, primary of weights $(2,0)$. Since we will use it only in the case where the internal transformation is abelian with one generator, we will remove the adjoint index like I. An important property is that ${\mathcal{𝒱}}_D$ is a real linear superfield

The reality is nothing but the reality condition $\nabla \mathcal{𝒲}=\overline{\nabla \mathcal{𝒲}}$. By the linearity (73), the nontrivial components are and the others are determined by them or vanish. Explicitly, the component fields are given by:

$${\mathcal{𝒱}}_D|=2iD,$$(75)

$${\nabla }_{\alpha }{\mathcal{𝒱}}_D|=-2i{\nabla }_{\alpha \dot{\alpha }}\overline{\lambda }{}^{\dot{\alpha }},$$(76)

$$\overline{\nabla }{}^{\dot{\alpha }}{\mathcal{𝒱}}_D|=-2i\overline{\nabla }{}^{\dot{\alpha }\alpha }{\lambda }_{\alpha },$$(77)

$$[{\nabla }_{\alpha },{\overline{\nabla }}_{\dot{\alpha }}]{\mathcal{𝒱}}_D|=4i{({\sigma }^b)}_{\alpha \dot{\alpha }}(e_a{}^m{\mathcal{𝒟}}_m-2b_a)F^{ba}+\mathrm{\text{fermions}},$$(78)

where $F_{ba}=e_b{}^n{e}_a{}^m{F}_{nm}$. A detailed form of (78) can be obtained by noticing that (78) can be written as : and then applying the formulae for the vector covariant derivatives in App. A. The definition of the covariant derivative ${\mathcal{𝒟}}_m$ is also given there (nothing but the ordinary Poincaré covariant derivative). The other components are given by:

$${\bar{\nabla }}^2{\nabla }_{\alpha }{\mathcal{𝒱}}_D|=4i{({\sigma }^a)}_{\alpha \dot{\alpha }}{\nabla }_a{\bar{\nabla }}^{\dot{\alpha }}{\mathcal{𝒱}}_D|,{\nabla }^2{\bar{\nabla }}^{\dot{\alpha }}{\mathcal{𝒱}}_D|=4i{({\bar{\sigma }}^a)}^{\dot{\alpha }\alpha }{\nabla }_a{\nabla }_{\alpha }{\mathcal{𝒱}}_D|,$$(80)

$${\nabla }^{\alpha }{\bar{\nabla }}^2{\nabla }_{\alpha }{\mathcal{𝒱}}_D|=-8{\nabla }^a{\nabla }_a{\mathcal{𝒱}}_D|.$$(81)

Gauge fixing to super Poincaré and compensators

The invariant integrals (37)–(39) are invariant under parallel transport and the origin-preserving superconformal transformations. On the other hand, our main target is super-Poincaré invariant actions.^b Therefore, we need a step of gauge-fixing the degrees of freedom of the dilatation, chiral $U(1)$ rotation and conformal translations.

A popular method of the gauge-fixing is to use so-called compensating superfields, or compensators. The point is to assign nontrivial weights to compensators so that it enables the desired gauge fixing.

Here we present a popular gauge-fixing procedure with chiral and anti-chiral compensators, which is associated with the old minimal supergravity.^c This consists of two steps. The first step is to fix the connection superfield for the gauged dilatation to zero

This exhausts the degrees of freedom of the conformal translations, and hence the connections $F_M$ for conformal translation are constrained. For example, combining (82) with the curvature constraints $R_{\underline{\alpha }\underline{\beta }}(D)=0$ gives It is then natural to introduce new superfields $\mathcal{ℛ},\overline{\mathcal{ℛ}},G_{\alpha \dot{\alpha }}$ by The other components of the $K$-connections $F_A{}^B$ are also fixed and expressed in $R,\overline{R},G_a$. See [Butter] for their explicit forms.

To exhaust the other gauge degrees of freedom, we introduce chiral compensators, which we denote by $C,\overline{C}$ and define as

Let us fix the chiral compensators $C,\bar{C}$ to where g is a real scalar primary superfield of weights $(0,0)$. The chirality conditions ${\overline{\nabla }}_{\dot{\alpha }}C={\nabla }_{\alpha }\overline{C}=0$, combined with (82), give constraints for the spinor connections $A_{\underline{\alpha }}$ for chiral $U(1)$ rotation since the compensators have nonvanishing chiral $U(1)$ weights $\pm 2∕3$. Solving them gives^d Next, the curvature constraint $\{{\nabla }_{\alpha },{\bar{\nabla }}_{\dot{\alpha }}\}C=-2i{\nabla }_{\alpha \dot{\alpha }}C$ allows us to express the vector connection $A_a$ in the spinor ones $A_{\underline{\alpha }}$. Using their fixed forms (87), we obtain Using them, we can obtain the component expansion of the chiral compensators : and

So far, we did not specify the detail of the function g. Therefore, the results here apply to any g. In this paper, we consider the case where g is a function of chiral and anti-chiral superfields of weights $(0,0)$.

2.2. Examples of actions

In this subsection, we shall present examples of supergravity actions that will be used later.

Pure super-Maxwell

The first example is an action of abelian gauge supermultiplet without compensators. Generalization to nonabelian case is straightforward. Actually, the discussion below goes in parallel with the globally supersymmetric case: by using the gaugino superfields and the F-type integral,

The action in component fields (component action) reads

In the presence of matter chiral superfields $\{{\mathrm{\Phi }}^i\}$ of vanishing weights, we are free to multiply a holomorphic function of them in front of ${\mathcal{𝒲}}^2$, called the gauge kinetic function,

The component action gets modified to where $h_R,h_I$ are the real, imaginary parts of $h(\varphi )$ (where ${\varphi }^i={\mathrm{\Phi }}^i|$) :

Actions for chiral matter superfields

Let us consider supergravity systems coupled with matter chiral and anti-chiral superfields $\{{\mathrm{\Phi }}^i\}$, $\{\bar{\mathrm{\Phi }}{}^{ī}\}$ of weights $(0,0)$. We recall that the invariant integrals (37)–(39) require the integrands $𝕍,𝕎$ to have the nontrivial weights, while the matter chiral sueprfields have vanishing weights. To remedy this gap, we can use the chiral compensators. Taking this into account, we can construct the following action :

K is called a Kähler potential, a real primary superfield of weights $(0,0)$, and W is called a superpotential, which is a chiral primary superfield of weights $(0,0)$ and a holomorphic function of $\{{\mathrm{\Phi }}^i\}$.

In principle, we can choose any gauge-fixing $C=\overline{C}=e^g$. However, there is a particular choice which provides the canonical normalization of the kinetic terms of the graviton and gravitino, namely the Einstein frame:^e

Under this, the bosonic part of the action (96) reads^f where we introduced

$$u=\mathcal{ℛ}|,ū=\overline{\mathcal{ℛ}}|,C=C|,\overline{C}=\overline{C}|,$$(99)

$${\varphi }^i={\mathrm{\Phi }}^i|,{\bar{\varphi }}^{ī}=\bar{\mathrm{\Phi }}{}^{ī}|,f^i=-\frac{1}{4}{\nabla }^2{\mathrm{\Phi }}^i|,{\bar{f}}^{ī}=-\frac{1}{4}\overline{\nabla }{}^2\bar{\mathrm{\Phi }}{}^{ī}|.$$(100)

The matrix $g_{i\bar{j}}:=K_{i\bar{j}}$ is called the Kähler metric. The differential operators ${\mathcal{𝒟}}_i,{\overline{\mathcal{𝒟}}}_{ī}$ on the superpotentials are defined by The field d is a linear combination of the internal gauge transformations of $C=C|$ and ${\varphi }^i$ : For example, when $C$ is neutral and ${\mathrm{\Phi }}^i$ is charged as $T{\mathrm{\Phi }}^i=-i{q}_i{\mathrm{\Phi }}^i$, we obtain $d=-i{\sum }_i{K}_i{q}_i{\varphi }^i$. We will later deal with the case where $C$ is also charged.

As a total action with the kinetic term of the abelian gauge multiplet, let us consider the model (96) plus (93).

We can see its component actions that $u,f^i,D$ are auxiliary fields. Solving their equations of motion for the auxiliary fields, we find that the scalar potential of this system reads

$$V=V_F+V_D,$$(104)

$$V_F=e^K(g^{\bar{j}i}{\mathcal{𝒟}}_{i}W{\mathcal{𝒟}}_{ī}\overline{W}-3W\overline{W}),$$(105)

$$V_D=\frac{1}{2}{h}_R^{-1}{d}^2,$$(106)

where $g^{\bar{j}i}$ is the inverse matrix of $g_{i\bar{j}}$. $V_F$ is called the F-term potential and $V_D$ the D-term potential. The first term in the bracket in $V_F$ comes from $f^i$ and the second from $\mathcal{ℛ}$.

Kähler transformation

We can see from the action (96) that it is invariant under the following transformation:

This is called Kähler transformation. This invariance implies that setting $J=lnW$ gives a Kähler invariant combination $\mathcal{𝒢}$ and in this Kähler frame the action (96) becomes

3. Fayet–Iliopoulos Terms

In this section, we review some aspects of Fayet–Iliopoulos terms in supergravity. In what follows, we only consider the case where the internal gauge transformation is $U(1)$, so we denote its generator by $T$. Adaptation to other abelian transformations is straightforward.

Let us consider the action (103). In the last section, we just consider the case where the chiral compensators are neutral under $T$. This case appears naturally when the superpotential is neutral under $T$ since the F-type integrals are required to be gauge invariant. On the other hand, we can consider the case where the superpotential transforms under $T$. In this case, the chiral compensators also have to transform under $T$. Such a gauged internal transformation is called gauged R transformation. Since we are considering gauged $U(1)$, we call it $U{(1)}_R$.

An important remark is that this characterization of the R-type transformation is sensitive to the Kähler transformation. Indeed, if we choose the Kähler frame in which the superpotential is just a constant, as mentioned above (110), the gauged $U(1)$ is not of R-type because the constant superpotential does not transform.

As we shall show shortly, the gauged $U{(1)}_R$ generates a Fayet–Iliopoulos (FI) term in the D-term scalar potential with an FI constant. This can be interpreted as an uplift of the vacuum energy. However, the FI constant is fixed by the charge of the superpotential under $U{(1)}_R$ and in this sense it is not completely a free parameter.

In 2017,¹ proposed a novel way to generate an FI term with a gauged $U(1)$ that is not of R-type. At the price of using the nonR transformation, the proposed action is highly nonlocal, while as we shall demonstrate later, the bosonic part of the action is local and the same as the ordinary bosonic action with an FI parameter. Furthermore, the FI constant is a completely free parameter.

The model in Ref. 1 has only the gravity and $U(1)$ gauge multiplets. Inspired by this, there appeared various generalizations and applications. Among them we would like to present our contributions^2,3,4 which constructed models that accommodate matter superfields with the new FI terms.

We shall first review the FI term from a gauged $U{(1)}_R$, and then move on to the model proposed by Ref. 1 and its generalizations with matter chiral superfields.

3.1. Fayet–Iliopoulos terms from gauged $U{(1)}_R$

We work with the model (103) with the matter chiral superfields charged under a gauged $U(1)$ as

Let us suppose that a Kähler potential is invariant under $U(1)$ while a superpotential transforms under the gauged $U(1)$ with charge Q, To maintain the gauge invariance of the F-type integrals, the combination $C^{3}W$ must be $T$-invariant, which forces $C$ to have charge $Q∕3$, Substituting (112) and (114) to (102), we find $d=-i({\sum }_i{q}_i{\mathrm{\Phi }}^i{K}_i+Q)$ and hence the D-term potential (106) becomes Here Q is the FI parameter.

Several models of this type were proposed in, for example,^5,6,7 in which vacua with tuneable vacuum energy, including a positive one (de Sitter vacuum), are realized by balancing the F-term and D-term potentials.

Note that since Q is the $U{(1)}_R$ charge of the superpotential, the FI parameter is constrained by the superpotential.

$U{(1)}_R$ charges after gauging fixing to super-Poincaré SUGRA

The chiral compensators are charged under $U{(1)}_R$, and hence its gauge fixing breaks the  $U{(1)}_R$ invariance in the resulting super-Poincaré theory.

However, there is a mixture of the $U{(1)}_R$ and the chiral $U(1)$ rotation that keeps the fixed compensators invariant. This can be interpreted as a new $U{(1)}_R$ in the gauge fixed theory with super Poincaré invariance.

To be concrete, let us note that the chiral compensator $C$ is charged under $U{(1)}_R$ and the chiral $U(1)$ rotation as

We can then easily see that the following combination of the generators : keeps the chiral compensators invariant, and hence it survives the gauge fixing. In this way, the new gauged $U{(1)}_R$ invariance is realized in the gauged-fixed theory with super-Poincaré invariance.

Let us compute the $R$-charges of the component fields in our model (103). Since spinor covariant derivatives are charged under $A$, different component fields have different $R$ charges. For example, if ${\mathrm{\Phi }}^i$ is charged as $T{\mathrm{\Phi }}^i=-i{q}_i{\mathrm{\Phi }}_i$, its component fields are charged under $R$ as

For the gaugino, using the weights $(3∕2,1)$ of ${\mathcal{𝒲}}_{\alpha }$, we obtain While the gravitino is neutral under the original $U{(1)}_R$, it transforms under $A$ and hence under $R$. To see this, we need to compute ${\delta }_A{E}_m{}^{\underline{\alpha }}$ since ${\psi }_m{}^{\underline{\alpha }}=2E_m{}^{\underline{\alpha }}|$, We therefore obtain

In general, the $U{(1)}_R$ is anomalous. It must be cancelled because the $U{(1)}_R$ is gauged. We will not give any detail, but¹² proposed an anomaly cancellation mechanism which uses a nontrivial gauge kinetic function of the form (where we assume that the theory has only one matter chiral superfield),

Under $U{(1)}_R$, it generates a term of the form $\beta F\wedge F$, and it was shown that an appropriate choice of $\beta$ achieves the anomaly cancellation (Green–Schwarz mechanism).

3.2. New FI terms

We demonstrated how the FI term appears in the presence of a gauged $U{(1)}_R$ transformation, and pointed out the property that the FI constant is constrained by the $U{(1)}_R$ charge of the superpotential.

In contrast, the action proposed in Ref. 1 has a gauged internal $U(1)$ but it is not of R-type. Furthermore, the FI parameter allows an arbitrary uplift of the vacuum energy.

For other constructions and reviews of new FI terms, see, for example, Refs. 13–19.

Without matter superfields

We first introduce the action.¹ The invariant integral that gives the new FI term is given by

where the internal gauge transformation is $U(1)$, with respect to which $\mathcal{𝒲}$ and $V_D$ are defined. Though the action is nonlocal in the superfields, the bosonic part of the action turns out to be quite simple : where we denoted $C|=c$.

Let us complete the action by adding the super-Maxwell action :

Gauge-fixing $c=\bar{c}=1$ gives the action in the Einstein frame. Solving the of motion for D that is auxiliary, we obtain the potential While D appears just linearly in the bosonic part, we can show that the fermionic part becomes singular if $D=0$. Therefore, the supersymmetry must be broken with a nonzero vacuum expectation value of D (D-term supersymmetry breaking). The goldstino is then the gaugino.

With matter chiral superfields – not Kähler invariant

A natural generalization of the action in Ref. 1 is to couple it with matter superfields. In Ref. 2, such a generalization was proposed.

A natural coupling of (125) with matter chiral superfields $\{{\mathrm{\Phi }}^i\}$ that are charged as $T{\mathrm{\Phi }}^i=-i{q}_i{\mathrm{\Phi }}^i$ may be

Here we assume that the Kähler potential and superpotential are invariant and thus the $U(1)$ is not of R-type. Under the Einstein-frame gauge fixing $C=\overline{C}=e^{K∕6}$, the bosonic part of $S_{\mathrm{\text{FI}}}^0$ becomes The D-term scalar potential contains a field-dependent FI term : This expression, as well as the bosonic part (128), indicates that the theory (127) is not Kähler invariant. This can be shown more directly as follows. Under the Kähler transformation with $C\mapsto e^{J∕3}C$, the factor ${\bar{w}}^2$ transforms as ${\bar{w}}^2\mapsto e^{-2\bar{J}∕3}{\bar{w}}^2$. If we want $S_{\mathrm{\text{FI}}}$ to be Kähler invariant, we need $T(e^{-2\bar{J}∕3}{\bar{w}}^2)=e^{-\bar{J}∕3}T({\bar{w}}^2)$. But this does not hold. Therefore, the new FI action $S_{\mathrm{\text{FI}}}$ violates the Kähler invariance.

Let us consider whether the FI terms of the two types, one in 3.1 and the other being (128), can be accommodated together or not. The answer is no as long as the gauged abelian transformation is of R-type. This is because the compensators in (128) get rescaled under the transformation and hence (128) cannot be invariant for the same reason as the argument above for the Kähler noninvariance.

Nevertheless, the answer becomes yes if we work in the Kähler frame in which the superpotential is not charged under $U(1)$. For example, let us consider a theory with a single chiral superfield $\mathrm{\Phi }$ with a Kähler potential K that is neutral $TK=0$, and the following superpotential,

where a is a constant. The gauged $U(1)$ is of R-type if $\xi$ does not vanish. Let us Kähler-transform this pair into Note that $T{K}^{\prime }$ is neutral too. The action is then plus the super-Maxwell action (91). The D-term scalar potential becomes Indeed, it contains two FI terms: the first one has the same as in the case with a gauged $U{(1)}_R$, and the second is from $S_{\mathrm{\text{FI}}}^0$. We will introduce applications of the scalar potential to inflation in Subsec. 4.2.

With matter chiral superfields — Kähler invariant

A natural goal that comes next is to search for Kähler-invariant generalizations of the new FI term in the presence of matter superfields. A crucial point is to work with Kähler invariant combinations of the superfields involved. For example, we should use $C\overline{C}{e}^{-K∕3}$ instead of $C\overline{C}$. We also need a Kähler invariant generalization of $\overline{w}{}^2$ such that its chiral projection becomes primary. It was found in Ref. 3 that the following superfield :

satisfies these requirements. It has weights $(-1,-2)$, and hence the chiral projection $\overline{T}({\bar{w}}_K^2)$ is primary with weights $(0,0)$. We can then construct the following D-type invariant action : This reduces to $S_{\mathrm{\text{FI}}}^0$ if K is set to zero. The action is consistent for any choice of the parameter $\xi$.

As in the case without matter, the bosonic part of this action is quite simple :

For example, let us consider the following model : plus the super-Maxwell action (91). Choosing $c=\bar{c}=e^{K∕6}$ for the Einstein frame, the D-term potential reads Indeed, this depends on K and W only through the Kähler-invariant combination $K+lnW$.

The action with two fermions was also computed in Ref. 3. As in the case without matter chiral superfields, the action becomes singular if the D-auxiliary field is zero.

Another Kähler invariant generalization

Another generalization of new FI terms with Kähler invariance has been explored by Antoniadis and Rondeau.⁴

Note first that $C\overline{C}{e}^{-K∕3}$ are not the only Kähler invariant quantity, but we have the following Kähler invariant combinations :

Taking into account the weights of the first two $(2,0)$ and $(1,2∕3)$, respectively,⁴ we introduced the following superfields : Since ${\bar{w}}_W^2$ has weights $(1,-2∕3)$, its chiral projection is primary of weights $(2,4∕3)$. We can then construct the following one-parameter family of the invariant D-type actions : The bosonic part becomes This can be interpreted as a field-dependent FI term. Setting $\alpha =-2∕3$ yields the bosonic part of $S_{\mathrm{\text{FI}}}^K$, while it is not clear whether its fermionic part becomes equal to that of $S_{\mathrm{\text{FI}}}^K$.

For example, let us consider the following model :

with the super-Maxwell action (91). Choosing $c=\bar{c}=e^{K∕6}$ for the Einstein frame, the D-term potential reads

4. Applications to Inflation

Building inflationary models in supergravity is challenging due to a number of issues, including fine-tuning to meet the slow-roll requirements, large field initial conditions that break the validity of the effective field theory and stabilization of the (pseudo) scalar companion of the inflaton arising from the fact that the number of bosonic components of superfields are always even. The simplest argument to see the fine tuning of the potential is that a canonically normalized kinetic term of a complex scalar field $\mathrm{\Phi }$ corresponds to a quadratic Kähler potential $K=\mathrm{\Phi }\bar{\mathrm{\Phi }}$ that brings one unit contribution to the slow-roll parameter $\eta =V^{\prime \prime }∕{\kappa }^{2}V$,^g arising from the proportionality factor $e^{{\kappa }^{2}K}$ in the expression of the scalar potential V. This problem can be avoided in models with no-scale structure where cancellations arise naturally due to noncanonical kinetic terms leading to potentials with flat directions (at the classical level). However, such models require often trans-Planckian initial conditions that invalidate the effective supergravity description during inflation. A concrete example where all these problems appear is the Starobinsky model of inflation, despite its phenomenological success.

In this section, we present two classes of models that avoid all three problems above in which the inflaton is identified with the superpartner of goldstino. The F-term in the scalar potential dominates the first class of models and drives inflation, while the D-term dominates the second. They are small field models where inflation takes place at a plateau around the maximum of the scalar potential (hill-top), and hence no large field initial conditions are required. The pseudo-scalar companion of the inflaton is absorbed by the gauge field that becomes massive, leaving the inflaton as a single scalar degree of freedom present in the low-energy spectrum. As we will show below, this model provides a minimal realization of natural small-field inflation in supergravity, compatible with present observations. Moreover, it allows a realistic minimum describing our present Universe with an infinitesimal positive vacuum energy without affecting the properties of the inflationary plateau.

4.1. F-term dominated inflation model

Let us consider the Kähler potential as a function of $\mathrm{\Phi }\bar{\mathrm{\Phi }}$ and expand the Kähler potential up to cubic terms order in $\mathrm{\Phi }\bar{\mathrm{\Phi }}$ while we choose the linear superpotential :

where A, B and $\tilde{f}$ are dimensionless parameter. One should think of the above form as a perturbative expansion around the canonical kinetic terms with coefficients less than unity. This model has $U{(1)}_R$ gauge symmetry and suppose that $\mathrm{\Phi }$ has a charge of q. With this, the total scalar potential $V:=V_F+V_D$ becomes Note that we define $\mathrm{\Phi }|=\rho e^{i\kappa \phi }$. The phase $\phi$ is absorbed by the $\mathrm{\text{U}}{(1)}_R$ gauge field that becomes massive away from the origin in the standard Brout–Englert–Higgs mechanism. In this model, the D-term has a constant FI contribution but plays no role during inflation.

However, in order to calculate the slow-roll parameters, we introduce the canonically normalized field $\chi$ satisfying

The slow-roll parameters can be defined in terms of the canonical field $\chi$ as

The number of e-folds N during inflation is determined by where we choose $|\eta ({\chi }_{\mathrm{\text{end}}})|=1$. The amplitude of density fluctuations ${\mathcal{𝒜}}_s$, the spectral index $n_s$ and the tensor-to-scalar ratio r can be written in terms of the slow-roll parameters : evaluated at the horizon exit. The Hubble parameter during inflation is approximate by $H_{\ast }=\kappa \sqrt{V_{\ast }∕3}$. Since inflation arises near the maximum $\kappa \rho =0$, we expand

$$ϵ=4{\left(\frac{-4A+y^2}{2+y^2}\right)}^2{(\kappa \rho )}^2+\mathcal{𝒪}({\rho }^4),$$(151)

$$\eta =2\left(\frac{-4A+y^2}{2+y^2}\right)+\mathcal{𝒪}({\rho }^2),$$(152)

where we defined $y=q∕\tilde{f}$. Notice that only the coefficient of the quadratic term A appears in the lowest order of slow-row parameters. The higher order coefficient will be important for tuning the cosmological constant at the minimum. The above equation implies $ϵ\simeq {\eta }^2{(\kappa \rho )}^2\ll \eta$. For simplicity, we focus on the special case $y\to 0$ where F-term contribution to the scalar potential is dominant. By considering the behavior near the origin, we can put some constraints on the coefficient A of the quadratic term of the Kähler potential. We can easily show that $A>0$ is required for having a local maximum of the scalar potential at $\rho =0$. Furthermore, the slow-roll condition $|\eta |\ll 1$ sets an upper bound $A\ll 0.25$. Taking these requirements into account, the constraint on A is To produce $\eta \sim -0.02$, we may choose $A\sim 0.005$, which is consistent with the CMB observational data. Note that a generalization version of the Fayet–Iliopoulos (FI) model was introduced in Ref. 20 as an example of the microscopic origin for the effective field theory of this class of inflation models. However, we need to turn on B and q in order to have a more realistic model with a small cosmological constant at the minimum of the potential. In this situation, A does not have to be very small. As a concrete example, we choose the following parameter values : which leads to the inflationary parameters $n_s=0.9543$, $r=1.72\times 10^{-6}$ and $H_{\ast }=3.25\times 10^{11}\mathrm{\text{GeV}}$. The above models are in agreement with cosmological observations²¹ and in the simplest case predict a rather small tensor-to-scalar ratio of primordial perturbations.

4.2. An example for D-term inflation model

The new FI-terms and their resulting D-term scalar potential discussed in the previous section provides a realization of inflation from supersymmetry breaking, driven by a D-term. The inflaton is chosen to be a superpartner of the goldstino, where its pseudoscalar partner is absorbed by the gauge field away from the origin. In this section, we work in the Kähler frame where the superpotential does not transform, and take into account the two types of FI terms which were discussed in Subsec. 3.2. After restoring the inverse reduced Planck mass $\kappa$, Eq. (131) can be written as

Let us choose the Kähler potential in canonical form : After performing a change of the field variable $\mathrm{\Phi }|=\rho e^{i\kappa \phi }$, the F-term contribution to the scalar potential is given by and the D-term contribution is Note that we set $b=\xi$ and rescaled the second FI parameter by $b^{\prime }={\xi }^{\prime }∕q$.

For $a=0$, one finds that for $b^{\prime }<-1$ and $b=3$, the potential has a maximum at the origin, and a supersymmetric minimum. Since the superpotential vanishs the SUSY breaking is measured by the D-term order parameter. Supersymmetry is broken at the local maximum and during inflation. On the other hand, at the global minimum, supersymmetry is preserved and the potential vanishes. Strictly speaking, the supersymmetric minimum is not valid because the new FI term becomes singular since the D-auxiliary vanishes. Therefore a small a is required in any case.

For $a\ne 0$, the potential still has a local maximum at $\rho =0$ for $b=3$ and $b^{\prime }<-1$. For this choice, the derivatives of the potential have the following properties :

For $b^{\prime }<-1$, the extremum at $\rho =0$ is a local maximum. As long as $a^2∕q^2\ll 1$, the change in the global minimum is very small, of order $\mathcal{𝒪}(a^2∕q^2)$. The plot of this change is shown in Fig. 1.

Let us comment on supersymmetry breaking. In the present case $a\ne 0$, the order parameters are both the Killing potential $\mathcal{𝒟}$ and the F-term contribution ${\mathcal{ℱ}}_X$, which read

Therefore, near the local maximum, ${\mathcal{ℱ}}_X∕\mathcal{𝒟}\sim \frac{a}{q}{\rho }^2$. On the other hand, at the global minimum, both $\mathcal{𝒟}$ and ${\mathcal{ℱ}}_X$ are of the same order i.e. ${\mathcal{ℱ}}_X∕\mathcal{𝒟}\sim \frac{a}{q}$, assuming that $\rho$ at the minimum is of order $\mathcal{𝒪}(1)$, which is true in the model we consider below. This makes tuning of the vacuum energy between the F- and D-contribution possible in principle.

Let us focus on the $b=3$ case and assume that the scalar potential is D-term dominated (fixing $a=0$), the model has only two free parameters, namely q and $b^{\prime }$. The first parameter controls the overall scale of the potential and it will be fixed by the amplitude ${\mathcal{𝒜}}_s$ of the CMB data.²¹ The only free-parameter left over is the second parameter $b^{\prime }$. Since we assume inflation to start near the origin $\rho =0$, the expansion of slow-roll parameters for small $\kappa \rho$ can be written as

Note also that $\eta$ is negative when $b^{\prime }<-1$. We can therefore tune the parameter $b^{\prime }$ to avoid the $\eta$-problem. The observation is that at $b^{\prime }=-1$, the effective charge of $\mathrm{\Phi }$ vanishes and thus the $\rho$-dependence in the D-term contribution (158) becomes of quartic order.

The number of e-folds N during inflation is determined by using Eq. (149). Note that the slow-roll parameters for small ${\rho }^2$ satisfy the simple relation $ϵ=\eta {(0)}^2{\rho }^2+O({\rho }^4)$ by Eq. (161). Therefore, the number of e-folds between $\rho ={\rho }_1$ and ${\rho }_2$ (${\rho }_1<{\rho }_2$) takes the following simple approximate form as in Ref. 5,

as long as the expansions in (161) are valid in the region ${\rho }_1\le \rho \le {\rho }_2$. Note that we used the approximation $\eta (0)\simeq {\eta }_{\ast }$, which holds in this case.

By using the power spectrum of scalar perturbations of the CMB such the amplitude ${\mathcal{𝒜}}_s$, tilt $n_s$, and the tensor-to-scalar ratio r, we are now comparing the theoretical predictions of this model to the observational data. From the relation of the spectral index above, one should have ${\eta }_{\ast }\simeq -0.02$, and thus Eq. (162) gives approximately the desired number of e-folds when the logarithm is of order one. Actually, using this formula, we can estimate the upper bound of the tensor-to-scalar ratio r and the Hubble scale $H_{\ast }$ following the same argument given in Ref. 5; the upper bounds are given by computing the parameters $r,H_{\ast }$ assuming that the expansions (161) hold until the end of inflation. We then get the bound

where we used ${\eta }_{\ast }=-0.02$, $N\simeq 50-60$ and ${\rho }_{\mathrm{\text{end}}}\lesssim 0.5$, which are consistent with our models.

A small field inflation model from supergravity with observable tensor-to-scalar ratio

Supergravity models with higher r are of particular interest. In this section, we demonstrate that our model can achieve large r, but only at the cost of adding a few additional terms to the Kähler potential. Let us consider the previous model with additional quadratic and cubic terms in $\mathrm{\Phi }\bar{\mathrm{\Phi }}$ :

while the superpotential remains constant as in Eq. (155). We assume that inflation is driven by the D-term by setting the parameter $a=0$. The scalar potential in terms of the field variable $\rho$ can be written as We now have two additional parameters A and B. These parameters do not affect our previous discussions on the choices of the parameter b because they appear in higher orders in $\rho$ in the scalar potential. Therefore, we can continue with the $b=3$ case. The formula (162) for the number of e-folds also holds for small ${(\kappa \rho )}^2$ even when $A,B$ are not zero because the new parameters appear at order ${\rho }^4$ and higher. However these two parameters can increase the value of the tensor-to-scalar ratio r. To obtain $r\approx 0.01$, we can choose for example By choosing the initial condition $\kappa {\rho }_{\ast }=0.445$ and $\kappa {\rho }_{\mathrm{\text{end}}}=1.155$, we get the results $N=58$, $n_s=0.96$, $r=0.01$ and ${\mathcal{𝒜}}_s=2.2\times 10^{-9}$, which is in agreement with the CMB data. Note that applications of the new FI term in no-scale supergravity model for inflation can also be found in literature works.^4,19,22

5. Conclusion

We summarized constructions of FI terms in supergravity, which were classified into two classes: one with R-type gauged abelian transformations, the other with nonR type ones. We then presented four models in the latter class: the first one does not have matter chiral superfields, the second one has chiral superfields with Kähler invariance violated, and the third and fourth ones have chiral superfields with Kähler invariance. In particular, we presented in the second model a scalar potential with two FI constants.

As applications of the scalar potential with two FI terms, we presented inflation models where inflation is driven by the D-term and the inflaton is the sgoldstino. The simplest case with the quadratic Kähler potential predicts a rather small tensor-to-scalar ratio of primordial perturbations, while the predictions with large tensor-to-scalar ratio can be obtained by introducing additional terms to the Kähler potential. It might be interesting to couple our D-term inflation model with the standard model. This can be done in the same way as presented in the author’s recent work.²³

Acknowledgments

This work was supported in part by the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation [grant numbers B01F650006 and B05F650021]. AC is also supported in part by Thailand Science research and Innovation Fund Chulalongkorn University (IND66230009).

ORCID

Ignatios Antoniadis  https://orcid.org/0000-0001-7521-3359

Auttakit Chatrabhuti  https://orcid.org/0000-0001-7353-3842

Hiroshi Isono  https://orcid.org/0000-0002-3473-2483

Notes

^a We use the boldface font for functions on superspace.

^b A popular exception is the superconformal action for Yang–Mills fields.

^c Another popular choice of compensators is a real linear superfield, associated with the new minimal supergravity.

^d The derivations of them as well as of the other expressions with covariant derivatives of $C$ below consist of three steps: First, we rewrite covariant derivatives on $C$ by using ${\nabla }_{A}C=E_A{}^M{\nabla }_{M}C$, the definition of ${\nabla }_M$, and the gauge fixing condition (82). Note that in this step, $C$ is kept unfixed. The resulting expressions are composed of $A_A$ and super-Poincaré covariant derivatives ${\mathcal{𝒟}}_A:=E_A{}^M({\partial }_M-\frac{1}{2}{\mathrm{\Omega }}_M{}^{ba}{M}_{ab})$ acting on $C$. The next step is to replace $C$ by its fixed form $e^g$. This yields expressions in which ${\mathcal{𝒟}}_A$’s act on g. Finally, we rewrite ${\mathcal{𝒟}}_A$’s on g in superconformal ones ${\nabla }_A$ on g, taking into account the vanishing weights of g and the condition (82).

^e This can be seen easily from the first line of (39).

^f Note that $G_{\alpha \dot{\alpha }}|$ does not appear. This is due to the particular gauge choice (97). This field appears as an auxiliary field in the fermionic part and contributes to the four-fermion interactions after it is integrated out.

^g Here $\kappa$ is the inverse of the reduced Planck mass, ${\kappa }^{-1}=2.4\times 10^{15}$TeV.