# Fayet–Iliopoulos terms in supergravity and D-term inflation

## Abstract

We analyse the consequences of a new gauge invariant Fayet–Iliopoulos (FI) term proposed recently to a class of inflation models driven by supersymmetry breaking with the inflaton being the superpartner of the goldstino. We first show that charged matter fields can be consistently added with the new term, as well as the standard FI term in supergravity in a Kähler frame where the *U*(1) is not an R-symmetry. We then show that the slow-roll conditions can be easily satisfied with inflation driven by a D-term depending on the two FI parameters. Inflation starts at initial conditions around the maximum of the potential where the *U*(1) symmetry is restored and stops when the inflaton rolls down to the minimum describing the present phase of our Universe. The resulting tensor-to-scalar ratio of primordial perturbations can be even at observable values in the presence of higher order terms in the Kähler potential.

## 1 Introduction

In a recent work [1], we proposed a class of minimal inflation models in supergravity that solve the \(\eta \)-problem in a natural way by identifying the inflaton with the goldstino superpartner in the presence of a gauged R-symmetry. The goldstino/inflaton superfield has then charge one, the superpotential is linear and the scalar potential has a maximum at the origin with a curvature fixed by the quartic correction to the Kähler potential *K* expanded around the symmetric point. The D-term has a constant Fayet–Iliopoulos (FI) contribution but plays no role in inflation and can be neglected, while the pseudoscalar partner of the inflaton is absorbed by the \(U(1)_R\) gauge field that becomes massive away from the origin.

Recently, a new FI term was proposed [2] that has three important properties: (1) it is manifestly gauge invariant already at the Lagrangian level; (2) it is associated to a *U*(1) that should not gauge an R-symmetry and (3) supersymmetry is broken by (at least) a D-auxiliary expectation value and the extra bosonic part of the action is reduced in the unitary gauge to a constant FI contribution leading to a positive shift of the scalar potential, in the absence of matter fields. In the presence of neutral matter fields, the FI contribution to the D-term acquires a special field dependence \(e^{2K/3}\) that violates invariance under Kähler transformations.

In this work, we study the properties of the new FI term and explore its consequences to the class of inflation models we introduced in [1].^{1} We first show that matter fields charged under the *U*(1) gauge symmetry can consistently be added in the presence of the new FI term, as well as a non-trivial gauge kinetic function. We then observe that the new FI term is not invariant under Kähler transformations. On the other hand, a gauged R-symmetry in ordinary Kähler invariant supergravity can always be reduced to an ordinary (non-R) *U*(1) by a Kähler transformation. By then going to such a frame, we find that the two FI contributions to the *U*(1) D-term can coexist, leading to a novel contribution to the scalar potential.

The resulting D-term scalar potential provides an alternative realisation of inflation from supersymmetry breaking, driven by a D- instead of an F-term. The inflaton is still a superpartner of the goldstino which is now a gaugino within a massive vector multiplet, where again the pseudoscalar partner is absorbed by the gauge field away from the origin. For a particular choice of the inflaton charge, the scalar potential has a maximum at the origin where inflation occurs and a supersymmetric minimum at zero energy, in the limit of negligible F-term contribution (such as in the absence of superpotential). The slow roll conditions are automatically satisfied near the point where the new FI term cancels the charge of the inflaton, leading to higher than quadratic contributions due to its non trivial field dependence.

The Kähler potential can be canonical, modulo the Kähler transformation that takes it to the non R-symmetry frame. In the presence of a small superpotential, the inflation is practically unchanged and driven by the D-term, as before. However, the maximum is now slightly shifted away from the origin and the minimum has a small non-vanishing positive vacuum energy, where supersymmetry is broken by both F- and D-auxiliary expectation values of similar magnitude. The model predicts in general small primordial gravitational waves with a tensor-to-scaler ration *r* well below the observability limit. However, when higher order terms are included in the Kähler potential, one finds that *r* can increase to large values \(r\simeq 0.015\).

The outline of our paper is the following. In Sect. 2, we review the new FI term (Sect. 2.1) and we show that matter fields charged under the *U*(1) gauge symmetry can consistently be added, as well as a non-trivial gauge kinetic function (Sect. 2.2). We also find that besides the new FI term, the usual (constant) FI contribution to the D-term [4] can also be present. Next, we show that the new FI term breaks the Kähler invariance of the theory, and therefore forbids the presence of any gauged R-symmetries. As a result, the two FI terms can only coexist in the Kähler frame where the *U*(1) is not an R-symmetry (Sect. 2.3). In Sect. 3, we compute the resulting scalar potential and analyse its extrema and supersymmetry breaking in both cases of absence (Sect. 3.1) or presence (Sect. 3.2) of superpotential. In Sect. 4, we analyse the consequences of the new term in the models of inflation driven by supersymmetry breaking. We first consider a canonical Kähler potential (Sect. 4.1) and then present a model predicting sizeable spectrum of primordial tensor fluctuations by introducing higher order corrections (Sect. 4.2).

## 2 On the new FI term

In this section we follow the conventions of [5] and set the Planck mass to 1.

### 2.1 Review

^{2}

*T*(\(\bar{T}\)) is defined in [7, 8], and leads to a chiral (antichiral) multiplet. For example, the chiral multiplet \(T(\bar{w}^2)\) has weights (2, 2). In global supersymmetry the operator

*T*corresponds to the usual chiral projection operator \(\bar{D}^2\).

^{3}

*a*,

*b*and coordinate indices \(\mu , \nu \). The fields \(w_\mu ^{ab}\), \(b_\mu \), and \(\mathcal A_\mu \) are the gauge fields corresponding to Lorentz transformations, dilatations, and \(T_R\) symmetry of the conformal algebra respectively, while \(\psi _\mu \) is the gravitino. The conformal d’Alembertian is given by \(\Box ^C = \eta ^{ab} \mathcal{D}_a \mathcal{D}_b \).

It is important to note that the FI term given by Eq. (1) does not require the gauging of an R-symmetry, but breaks invariance under Kähler transformations. In fact, we will show below that a gauged R-symmetry would forbid such a term \(\mathcal L_{FI}\).^{4}

*D*contains a term

### 2.2 Adding (charged) matter fields

*U*(1). For simplicity, we focus on a single chiral multiplet

*X*. The extension to more chiral multiplets is trivial. The Lagrangian is given by

*W*(

*X*) and a gauge kinetic function

*f*(

*X*). The first three terms in Eq. (8) give the usual supergravity Lagrangian [5]. We assume that the multiplet

*X*transforms under the

*U*(1),

*U*(1) is not an R-symmetry. In other words, we assume that the superpotential does not transform under the gauge symmetry. The reason for this will be discussed in Sect. 2.3. For a model with a single chiral multiplet this implies that the superpotential is constant

*f*(

*X*) is the gauge kinetic function. The F-term contribution to the scalar potential remains the usual

^{5}However, in the Kähler frame of Eq. (17) the superpotential transforms nontrivially under the gauge symmetry. As a consequence, the gauge symmetry becomes an R-symmetry. We will argue in Sect. 2.3 that

- 1.
The extra term (1) violates the Kähler invariance of the theory, and the two models related by a Kähler transformation are no longer equivalent.

- 2.
The model written in the Kähler frame where the gauge symmetry becomes an R-symmetry in Eq. (17) can not be consistently coupled to \(\mathcal L_{\text {FI}}\).

### 2.3 Kähler invariance and R-symmetry

*J*. In other words, the chiral compensator field \(S_0\) transforms under a Kähler transformation as

^{6}

As a result, a Kähler transformation is no longer a symmetry of the Lagrangian. Therefore, when the term \(\mathcal L_{\text {FI}}\) is included, two models that are related by a Kähler transformation are no longer (classically) equivalent. Nevertheless, it is still possible to add the usual FI term \(\xi _1\) in the Kähler frame where the gauge symmetry is not an R-symmetry.

*U*(1) gauge symmetry under which the superpotential transforms with a phase. For example, under the gauge transformations Eq. (9) the superpotential in Eq. (17) transforms as

^{7}

## 3 The scalar potential in a non R-symmetry frame

^{8}

*b*was introduced as a free parameter. We now proceed to narrowing the value of

*b*by the following physical requirements. We first consider the behaviour of the potential around \(\rho =0\),

*b*under the same requirement.

First, in order for \(\mathcal {V}(0)\) to be finite, we need \(b \ge 0\). We first consider the case \(b>0\). We next investigate the condition that the potential at \(\rho =0\) has a local maximum. For clarity we discuss below the cases of \(F=0\) and \(F \ne 0 \) separately. The \(b=0\) case will be treated at the end of this section.

### 3.1 Case \(F=0\)

*b*further, let us turn to the second derivative,

We conclude that for \(\xi < -1\) and \(b= 3/2\) the potential has a maximum at \(\rho = 0\), and a supersymmetric minimum at \(\rho _v\). We postpone the analysis of inflation near the maximum of the potential in Sect. 4, and the discussion of the uplifting of the minimum in order to obtain a small but positive cosmological constant below. In the next subsection we investigate the case \(F \ne 0\).

*q*and supersymmetry is broken. On the other hand, at the global minimum, supersymmetry is preserved and the potential vanishes.

### 3.2 Case \(F \ne 0\)

*b*further, let us turn to the second derivative,

*b*. Nevertheless, as we will show below, the potential can have a local maximum in the neighbourhood of \(\rho =0\) if we choose \(b=3/2\) and \(\xi < -1\). For this choice, the derivatives of the potential have the following properties,

*F*-contribution to the scalar potential small by taking \(F^2 \ll q^2|\xi +1|\), which guarantees the approximation ignoring higher order terms in \(\rho \). We now choose \(\xi <-1\) so that \(\rho \) for this extremum is positive. The second derivative at the extremum reads

A comment must be made here on the action in the presence of non-vanishing *F* and \(\xi \). As mentioned above, the supersymmetry is broken both by the gauge sector and by the matter sector. The associated goldstino therefore consists of a linear combination of the *U*(1) gaugino and the fermion in the matter chiral multiplet *X*. In the unitary gauge the goldstino is set to zero, so the gaugino is not vanishing anymore, and the action does not simplify as in Ref. [2]. This, however, only affects the part of the action with fermions, while the scalar potential does not change. This is why we nevertheless used the scalar potential (26) and (27).

Let us consider now the case \(b=0\) where only the new FI parameter \(\xi \) contributes to the potential. In this case, the condition for the local maximum of the scalar potential at \(\rho = 0\) can be satisfied for \(-\frac{3}{2}< \xi < 0\). When *F* is set to zero, the scalar potential (27) has a minimum at \(\rho _{\text {min}}^2 = \frac{3}{2}\ln \left( -\frac{3}{2\xi } \right) \). In order to have \(\mathcal {V}_{\text {min}} = 0\), we can choose \(\xi = - \frac{3}{2 e}\). However, we find that this choice of parameter \(\xi \) does not allow slow-roll inflation near the maximum of the scalar potential. Similarly to our previous models [1], it may be possible to achieve both the scalar potential satisfying slow-roll conditions and a small cosmological constant at the minimum by adding correction terms to the Kähler Potential and turning on a parameter *F*. However, in this paper, we will focus on \(b = 3/2\) case where, as we will see shortly, less parameters are required to satisfy the observational constraints.

## 4 Application in inflation

In the previous work [1], we proposed a class of supergravity models for small field inflation in which the inflation is identified with the sgoldstino, carrying a *U*(1) charge under a gauged R-symmetry. In these models, inflation occurs around the maximum of the scalar potential, where the *U*(1) symmetry is restored, with the inflaton rolling down towards the electroweak minimum. These models also avoid the so-called \(\eta \)-problem in supergravity by taking a linear superpotential, \(W \propto X\). In contrast, in the present paper we construct models with two FI parameters \(b,\xi \) in the Kähler frame where the *U*(1) gauge symmetry is not an R-symmetry. If the new FI term \(\xi \) is zero, our models are Kähler equivalent to those with a linear superpotential in [1] (Case 1 models with \(b=1\)). The presence of non-vanishing \(\xi \), however, breaks the Kähler invariance as shown in Sect. 2. Moreover, the FI parameter *b* cannot be 1 but is forced to be \(b=3/2\), according to the argument in Sect. 3. So the new models do not seem to avoid the \(\eta \)-problem. Nevertheless, we will show below that this is not the case and the new models with \(b = 3/2\) avoid the \(\eta \)-problem thanks to the other FI parameter \(\xi \) which is chosen near the value at which the effective charge of *X* vanishes between the two FI-terms. Inflation is again driven from supersymmetry breaking but from a D-term rather than an F-term as we had before.

### 4.1 Example for slow-roll D-term inflation

In this section we focus on the case where \(b = 3/2\) and derive the condition that leads to slow-roll inflation scenarios, where the start of inflation (or, horizon crossing) is near the maximum of the potential at \(\rho =0\). We also assume that the scalar potential is D-term dominated by choosing \(F =0\), for which the model has only two parameters, namely *q* and \(\xi \). The parameter *q* controls the overall scale of the potential and it will be fixed by the amplitude \(A_s\) of the CMB data. The only free-parameter left over is \(\xi \), which can be tuned to satisfy the slow-roll condition.

*X*vanishes and thus the \(\rho \)-dependence in the D-term contribution (27) becomes of quartic order.

*N*during inflation is determined by

*r*. These are written in terms of the slow-roll parameters:

*r*and the Hubble scale \(H_*\) following the same argument given in [1]; that is, the upper bounds are given by computing the parameters \(r,H_*\) assuming that the expansions (47) hold until the end of inflation. We then get the bound

The theoretical predictions for \(\rho _{*} = 0.055\) and \(\rho _\mathrm{end} = 0.403 \) and the parameters given in Eq. (52)

| \(n_s\) | | \(A_s\) |
---|---|---|---|

58 | 0.9541 | \(7.06 \times 10^{-6} \) | \(2.22 \times 10^{-9} \) |

As was shown in Sect. 3.1, this model has a supersymmetric minimum with zero cosmological constant because *F* is chosen to be zero. One possible way to generate a non-zero cosmological constant at the minimum is to turn on the superpotential \(W = \kappa ^{-3} F \ne 0\), as mentioned in Sect. 3.2. In this case, the scale of the cosmological constant is of order \(\mathcal {O}(F^2)\). It would be interesting to find an inflationary model which has a minimum at a tiny tuneable vacuum energy with a supersymmetry breaking scale consistent with the low energy particle physics.

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

*r*set by Planck, supergravity models with higher

*r*are of particular interest. In this section we show that our model can get large

*r*at the price of introducing some additional terms in the Kähler potential. Let us consider the previous model with additional quadratic and cubic terms in \(X\bar{X}\):

*A*and

*B*. This does not affect the arguments of the choices of

*b*in the previous sections because these parameters appear in higher orders in \(\rho \) in the scalar potential. So, we consider the case \(b = {3}/{2}\). The simple formula (49) for the number of e-folds for small \(\rho ^2\) also holds even when

*A*,

*B*are turned on because the new parameters appear at order \(\rho ^4\) and higher. To obtain \(r \approx 0.01\), we can choose for example

The theoretical predictions for \(\rho _{*} = 0.055\) and \(\rho _\mathrm{end} = 0.403 \) and the parameters given in Fig. 5

| \(n_s\) | | \(A_s\) |
---|---|---|---|

57 | 0.9603 | 0.015 | \(2.22 \times 10^{-9} \) |

In summary, in contrast to the model in [1] where the F-term contribution is dominant during inflation, here inflation is driven purely by a D-term. Moreover, a canonical Kähler potential (24) together with two FI-parameters (*q* and \(\xi \)) is enough to satisfy Planck’15 constraints, and no higher order correction to the Kähler potential is needed. However, to obtain a larger tensor-to-scalar ratio, we need to introduce perturbative corrections to the Kähler potential up to cubic order in \(X\bar{X}\) (i.e. up to order \(\rho ^6\)). This model provides a supersymmetric extension of the model [14], which realises large *r* at small field inflation without referring to supersymmetry.

## 5 Conclusions

In this paper, we have shown that charged matter fields can be consistently coupled with the recently proposed FI-term [2] in the frame where the superpotential is invariant under the *U*(1) symmetry. We demonstrated that Kähler transformations do not give equivalent theories. It would be interesting to explore the possibility of recovering Kähler covariance but obtaining the same physical action [10].

We then explored the possibility of obtaining inflation models driven by a D-term in the presence of the two FI terms. We first constrained one of the FI parameters by requiring that a slow-roll small-field inflation starts around the origin of the scalar potential which should be a local maximum. In the case where the superpotential vanishes, the potential has a global minimum preserving supersymmetry. We found explicit models in which the slow-roll conditions are satisfied and inflation is driven by the D-term. Although the predicted tensor-to-scalar ratio of primordial perturbations is quite small for canonical Kähler potential, we found that by adding perturbative corrections, we can achieve significantly larger ratios that could be observed in the near future.

These models provide an alternative realisation of inflation driven by supersymmetry breaking identifying the inflaton with the goldstino superpartner [1], but based on a D-term instead of an F-term.

We also discussed the case where the superpotential is turned on. Then, supersymmetry is broken at the global minimum but the supersymmetry breaking scale is of the order of the cosmological constant. In order to connect our model with low energy particle physics, one needs to find a mechanism for reconciling the hierarchy between the two scales in our model.

## Footnotes

- 1.
This new FI term was also studied in [3] to remove an instability from inflation in Polonyi-Starobinsky supergravity.

- 2.
A similar, but not identical term was studied in [6].

- 3.
The operator

*T*indeed has the property that \(T(Z) = 0\) for a chiral multiplet*Z*. Moreover, for a vector multiplet*V*we have \(T(Z C) = Z T(C)\), and \([C]_D = \frac{1}{2} [T(C) ]_F\). - 4.
We kept the notation of [2]. Note that in this notation the field strength superfield \(\mathcal W_\alpha \) is given by \(\mathcal W ^2 = \bar{\lambda }P_L \lambda \), and \((V)_D\) corresponds to \(\mathcal D^\alpha \mathcal W_\alpha \).

- 5.
At the quantum level, a Kähler transformation also introduces a change in the gauge kinetic function

*f*, see for example [9]. - 6.
- 7.
Note that this is technically not yet an R-symmetry: after fixing the conformal gauge, a mixture of the \(U(1)_R\) described above and the T-symmetry in the superconformal algebra is broken down to the usual R-symmetry in supergravity \(U(1)_R'\): \(U(1)_R \times U(1)_T \rightarrow U(1)_R'\).

- 8.
Strictly speaking, the gauge kinetic function gets a field-dependent correction proportional to \(q^2\ln \rho \), in order to cancel the chiral anomalies [1]. However, the correction turns out to be very small and can be neglected below, since the charge

*q*is chosen to be of order of \(10^{-5}\) or smaller.

## Notes

### Acknowledgements

This work was supported in part by the Swiss National Science Foundation, in part by a CNRS PICS grant and in part by the “CUniverse” research promotion project by Chulalongkorn University (grant reference CUAASC). The authors would like to thank Ramy Brustein, Toshifumi Noumi, Qaiser Shafi, and Antoine Van Proeyen for fruitful discussions.

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