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

Eur. Phys. J. C
Fayet-Iliopoulos terms in supergravity and D-term inflation
I. Antoniadis 1 2
A. Chatrabhuti 0
H. Isono 0
R. Knoops 0
0 Department of Physics, Faculty of Science, Chulalongkorn University , Phayathai Road, Pathumwan, Bangkok 10330 , Thailand
1 Albert Einstein Center, Institute for Theoretical Physics, University of Bern , Sidlerstrasse 5, 3012 Bern , Switzerland
2 Laboratoire de Physique Théorique et Hautes Energies - LPTHE, Sorbonne Université , CNRS, 4 Place Jussieu, 75005 Paris , France
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 Rsymmetry. 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 η-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
Dterm acquires a special field dependence e2K /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
1 This new FI term was also studied in [3] to remove an instability
from inflation in Polonyi-Starobinsky supergravity.
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 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
In [2], the authors propose a new contribution to the
supergravity Lagrangian of the form2
.
2 A similar, but not identical term was studied in [6].
The chiral compensator field S0, with Weyl and chiral
weights (Weyl, Chiral) = (
1, 1
), has components S0 =
(s0, PL 0, F0) . The vector multiplet has vanishing Weyl
and chiral weights, and its components are given by V =
v, ζ, H, vμ, λ, D . In the Wess-Zumino gauge, the first
components are put to zero v = ζ = H = 0. The
multiplet w2 is of weights (
1, 1
), and given by
w2 = λ¯ PS0L2 λ ,
w2
¯ =
λ PR2 λ¯ .
S¯0
The components of λ¯ PL λ are given by
λ¯ PL λ =
λ¯ PL λ ;
√
1
2 PL − 2 γ · Fˆ + i D λ ; 2 λ¯ PL D/ λ
+ Fˆ − · Fˆ − − D2 .
The kinetic terms for the gauge multiplet are given by
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
1
Lkin = − 4
λ¯ PL λ F + h.c. .
The operator T (T¯ ) is defined in [7,8], and leads to a
chiral (antichiral) multiplet. For example, the chiral multiplet
T (w¯ 2) has weights (
2, 2
). In global supersymmetry the
operator T corresponds to the usual chiral projection operator
D¯ 2.3
From now on, we will drop the notation of h.c. and
implicitly assume its presence for every [ ]F term in the Lagrangian.
Finally, the multiplet (V )D is a linear multiplet with weights
(
2, 0
), given by
(V )D =
D, D/ λ, 0, Db Fˆab, −D/ D/ λ, −
C D .
The definitions of D/ λ and the covariant field strength Fˆab
can be found in Eq. (17.1) of [5], which reduce for an abelian
gauge field to
Fˆab = eaμebν 2∂[μ Aν] + ψ¯ [μγν]λ
Dμλ =
∂μ − 2 bμ + 41 wμabγab − 2 i γ∗Aμ λ
3 3
− 41 γ ab Fˆab + 2 i γ∗ D
1
ψμ.
Here, eaμ is the vierbein, with frame indices a, b and
coordinate indices μ, ν. The fields wμab, bμ, and Aμ are the gauge
fields corresponding to Lorentz transformations, dilatations,
and TR symmetry of the conformal algebra respectively,
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 (ZC) =
Z T (C), and [C]D = 21 [T (C)]F .
while ψμ is the gravitino. The conformal d’Alembertian is
given by C = ηabDa Db.
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
LF I .4
The resulting Lagrangian after integrating out the
auxiliary field D contains a term
ξ 2
LFI,new = − 22 (s0s¯0)2 .
In the absence of additional matter fields, one can use the
Poincaré gauge s0 = s¯0 = 1, resulting in a constant D-term
contribution to the scalar potential. This prefactor however
is relevant when matter couplings are included in the next
section.
2.2 Adding (charged) matter fields
In this section we couple the term LFI given by Eq. (
1
) to
additional matter fields charged under the 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
Gauge invariance fixes the Kähler potential to be a function
of X eqV X¯ (for notational simplicity, in the following we omit
the eqV factors).
Indeed, in this case the term LFI can be consistently added
to the theory, similar to [2], and the resulting D-term
contribution to the scalar potential acquires an extra term proportional
to ξ2
1
VD = 2 Re ( f (X ))−1 i kX ∂X K + ξ2e 23 K 2 ,
(
7
)
where the Killing vector is kX = −i q X and f (X ) is the
gauge kinetic function. The F-term contribution to the scalar
potential remains the usual
L = −3 S0 S¯0e− 13 K (X,X¯ ) D + S03W (X )
F
1
− 4
f (X ) λ¯ PL λ F + LFI,
with a Kähler potential K (X, X¯ ), a superpotential 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
),
V → V +
X → X e−q ,
+ ¯ ,
with gauge multiplet parameter . We assume that the 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
W (X ) = F.
4 We kept the notation of [2]. Note that in this notation the field strength
superfield Wα is given by W2 = λ¯PL λ, and (V )D corresponds to
DαWα.
(
8
)
(
9
)
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(15)
(16)
(17)
VF = eK (X,X¯ )
−3W W¯ + g X X¯ ∇X W ∇¯ X¯ W¯ .
For a constant superpotential (
10
) this reduces to
VF = |F |2eK (X,X¯ )
−3 + g X X¯ ∂X K ∂X K .
¯
From Eq. (
11
) it can be seen that if the Kähler
potential includes a term proportional to ξ1 log(X X¯ ), the D-term
contribution to the scalar potential acquires another constant
contribution. For example, if
K (X, X¯ ) = X X¯ + ξ1 ln(X X¯ ),
the D-term contribution to the scalar potential becomes
1
VD = 2 Re ( f (X ))−1 q X X¯ + qξ1 + ξ2e 23 K 2 .
We will argue below that the contribution proportional to ξ1 is
the usual FI term in a non R-symmetric Kähler frame, which
can be consistently added to the model including the new FI
term proportional to ξ2.
In the absence of the extra term, a Kähler transformation
K (X, X¯ ) → K (X, X¯ ) + J (X ) + J¯(X¯ ),
W (X ) → W (X )e−J (X),
with J (X ) = −ξ1 ln X allows one to recast the model in the
form
K (X, X¯ ) = X X¯ ,
W (X ) = m3/2 X.
The two models result in the same Lagrangian, at least
classically.5 However, in the Kähler frame of Eq. (17) the
super5 At the quantum level, a Kähler transformation also introduces a
change in the gauge kinetic function f , see for example [9].
T¯ (w2)e− 23 J =
2 λ¯ PL D/ λ + F − · F − − D2
potential transforms nontrivially under the gauge
symmetry. As a consequence, the gauge symmetry becomes an
Rsymmetry. 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 LFI.
2.3 Kähler invariance and R-symmetry
In the superconformal compensator formalism of
supergravity the chiral compensator S0 is not uniquely defined: a
redefinition of the chiral compensator S0 = S0e J/3 results in a
Kähler transformation (16) with parameter J . In other words,
the chiral compensator field S0 transforms under a Kähler
transformation as
K (X, X¯ ) → K (X, X¯ ) + J (X ) + J¯(X¯ ),
W (X ) → W (X )e−J (X),
J(X)
S0 → S0e 3 .
Indeed, the first three terms in the Lagrangian (
8
) are invariant
under this transformation. However, LFI, repeated here for
convenience,
(18)
where the first component of T¯ (w2e− 23 J (X)) is given by (up
to 4-fermion terms)
T¯ (w2e− 23 J ) =
e−2 j/3
s02
− 2 λ¯PL λ
2√2
+ s0
2 λ¯PL D/ λ + F− · F− − D2
1 1
s0 F0 + 3 Fj
1 1
χ¯0 + 3 χ¯ j PL − 2 γ · F + i D λ
+ 4-fermions ; . . . ; . . . ,
(20)
and we used the notation J (X ) = ( j, PL χ j , Fj ). As a
comparison, the lowest component of T¯ (w2)e− 23 J does not
contain χ j and Fj , and is given by
e−2 j/3
s02
2
− s0 λ¯ PL λF0
2√2
+
χ¯0 PL
s0
1
− 2 γ · F + i D
λ
+ 4-fermions ; . . . ; . . . .
(21)
As a result T¯ (w2) is not Kähler covariant, and the term LFI
violates the Kähler invariance of the theory.6
As a result, a Kähler transformation is no longer a
symmetry of the Lagrangian. Therefore, when the term LFI 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 ξ1 in
the Kähler frame where the gauge symmetry is not an
Rsymmetry.
Indeed, a U (
1
)R -symmetry is a 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
W → W e−q ,
q
S0 → S0e 3 .
and the chiral compensator field transforms as7
(22)
(23)
Following the same arguments as above for the Kähler
transformations, one can see that the gauge invariance is violated
in LFI as a consequence of Eq. (23). As a result, in order
for the extra FI term to be consistently coupled to
supergravity, the superpotential should not transform. Although
the usual FI term in supergravity is usually associated with
a gauged R-symmetry [11,12], it is possible to rewrite a
theory with a non-zero superpotential (by a Kähler
transformation) in terms of a constant superpotential and the
Kähler potential given by K = K + ln(W W¯ ). This leads
to the same (classical) Lagrangian, while the gauge
symmetry that gave rise to the constant FI contribution is not
an R-symmetry. Therefore, in this ’Kähler frame’, the
theory can be consistently coupled to LFI. In fact, this was
the motivation behind the choice of the Kähler potential in
Eq. (
14
).
6 It would be interesting to see if this term can be modified in order to
keep Kähler invariance without modifying the resulting contribution to
the scalar potential Eq. (
7
), at least in the rigid limit [10].
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
Tsymmetry in the superconformal algebra is broken down to the usual
R-symmetry in supergravity U (
1
)R: U (
1
)R × U (
1
)T → U (
1
)R.
3 The scalar potential in a non R-symmetry frame
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 the last
section. For convenience, we repeat here the Kähler potential
in Eq. (
14
) and restore the inverse reduced Planck mass
κ = MP−l1 = (2.4 × 1018 GeV)−1:
K = κ−2(X X¯ + ξ1 ln X X¯ ).
The superpotential and the gauge kinetic function are set to
be constant:8
W = κ−3 F,
f (X ) = 1.
After performing a change of the field variable X = ρeiθ
where ρ ≥ 0 and setting ξ1 = b, the full scalar potential
V = VF + VD is a function of ρ. The F-term contribution to
the scalar potential is given by
1 F 2eρ2 ρ2b
VF = κ4
maximum at ρ = 0. In our last paper [1], we considered
models of this type with ξ = 0 (which were called Case 1
models), and found that the choice b = 1 is forced by the
requirement that the potential takes a finite value at the local
maximum ρ = 0. In this paper, we will investigate the effect
of the new FI parameter ξ on the choice of b under the same
requirement.
First, in order for V(0) to be finite, we need b ≥ 0. We first
consider the case b > 0. We next investigate the condition
that the potential at ρ = 0 has a local maximum. For clarity
we discuss below the cases of F = 0 and F = 0 separately.
The b = 0 case will be treated at the end of this section.
3.1 Case F = 0
In this case VF = 0 and the scalar potential is given by only
the D-term contribution V = VD. Let us first discuss the first
derivative of the potential:
q2
VD = 2κ4 4bρ 1 + O(ρ2) +
8b2
3
Note that we rescaled the second FI parameter by ξ = ξ2/q.
We consider the case with ξ = 0 because we are interested
in the role of the new FI-term in inflationary models driven
by supersymmetry breaking. Moreover, the limit ξ → 0 is
ill-defined [2].
The first FI parameter 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 ρ = 0,
b2 + 2bρ2 + O(ρ4) + 2bξρ 43b 1 + O(ρ2)
+ ξ 2ρ 83b 1 + O(ρ2) ,
ρ2b b2ρ−2 + (2b − 3) + O(ρ2) .
q2
VD = 2κ4
F 2
VF = κ4
In this paper we are interested in small-field inflation models
in which the inflation starts in the neighbourhood of a local
8 Strictly speaking, the gauge kinetic function gets a field-dependent
correction proportional to q2 ln ρ, 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.
(24)
(25)
(26)
(27)
(28)
(29)
For VD(0) to be convergent, we need b ≥ 3/4 (note that
ξ = 0). When b = 3/4, we have VD(0) = 8b2ξ /3, which
does not give an extremum because we chose ξ = 0. On
the other hand, when b > 3/4, we have VD(0) = 0. To
narrow the allowed value of b further, let us turn to the second
derivative,
q2
VD = 2κ4 4b 1 + O(ρ2) +
When 3/4 < b < 3/2, the second derivative VD(0) diverges.
When b > 3/2, the second derivative becomes VD(0) =
2κ−4q2b > 0, which gives a minimum.
We therefore conclude that to have a local maximum at
ρ = 0, we need to choose b = 3/2, for which we have
The condition that ρ = 0 is a local maximum requires ξ <
−1.
Let us next discuss the global minimum of the potential
with b = 3/2 and ξ < −1. The first derivative of the potential
without approximation reads
VD ∝ ρ(3 + 3ξ e 23 ρ2 + 2ξρ2e 23 ρ2 )(3 + 2ρ2 + 2ξρ2e 23 ρ2 ).
(30)
(31)
(32)
ξρ 43b −1 1 + O(ρ2)
(33)
Since 3 + 3ξ e 23 ρ2 + 2ξρ2e 23 ρ2 < 0 for ρ ≥ 0 and ξ < −1,
the extremum away from ρ = 0 is located at ρv satisfying
the condition
3 + 2ρv2 + 2ξρv2e 23 ρv2 = 0.
Substituting this condition into the potential VD gives
VD(ρv) = 0.
We conclude that for ξ < −1 and b = 3/2 the potential
has a maximum at ρ = 0, and a supersymmetric minimum at
ρv. 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 = 0.
We finally comment on supersymmetry (SUSY) breaking
in the scalar potential. Since the superpotential is zero, the
SUSY breaking is measured by the D-term order parameter,
namely the Killing potential associated with the gauged U(
1
),
which is defined by
D = i κ−2 −i q X
W
∂ W 2 ∂K W .
∂ X + κ ∂ X
This enters the scalar potential as VD = D2/2. So, at the local
maximum and during inflation D is of order q and
supersymmetry is broken. On the other hand, at the global minimum,
supersymmetry is preserved and the potential vanishes.
ρ
V
3.2 Case F = 0
VF = κ−4 F 2
In this section we take into account the effect of VF ; its first
derivative reads:
× b2(2b − 2)ρ2b−3 + 2b(2b − 3)ρ2b−1 1 + O(ρ2) .
For V (0) to be convergent, we need b ≥ 3/2, for which
VD(0) = 0 holds. For b = 3/2, we have VF (0) =
(9/4)κ−4 F 2 > 0, that does not give an extremum. For
b > 3/2, we have VF (0) = 0. To narrow the allowed values
of b further, let us turn to the second derivative,
VF = κ−4 F 2 b2(2b − 2)(2b − 3)ρ2b−4
+ 2b(2b − 3)(2b − 1)ρ2b−2 1 + O(ρ2) .
(37)
For 3/2 < b < 2, the second derivative VF (0) diverges. For
b ≥ 2, the second derivative is positive V (0) > 0, that gives
a minimum (note that VD(0) > 0 as well in this range).
3F 2
− 4q2(ξ + 1)
.
Note that the extremum is in the neighbourhood of ρ = 0
as long as we keep the F -contribution to the scalar
potential small by taking F 2 q2|ξ + 1|, which guarantees the
approximation ignoring higher order terms in ρ. We now
choose ξ < −1 so that ρ for this extremum is positive. The
second derivative at the extremum reads
(34)
(35)
(36)
(38)
(39)
(40)
(41)
We conclude that the potential cannot have a local
maximum at ρ = 0 for any choice of b. Nevertheless, as we
will show below, the potential can have a local maximum
in the neighbourhood of ρ = 0 if we choose b = 3/2 and
ξ < −1. For this choice, the derivatives of the potential have
the following properties,
V (0) < 0,
The extremisation condition around ρ = 0 becomes
3κ−4q2(ξ + 1)ρ + 49 κ−4 F 2
0.
So the extremum is at
3κ−4q2(ξ + 1),
as long as we ignore higher order terms in F 2/(q2|ξ + 1|).
By our choice ξ < −1, the extremum is a local maximum,
as desired.
Let us comment on the global minimum after turning on
the F-term contribution. As long as we choose the parameters
so that F 2/q2 1, the change in the global minimum ρv
is very small, of order O(F 2/q2), because the extremisation
condition depends only on the ratio F 2/q2. So the change in
the value of the global minimum is of order O(F 2). The plot
of this change is given in Fig. 1.
In the present case F = 0, the order parameters of SUSY
breaking are both the Killing potential D and the F-term
contribution FX , which read
D ∝ q( 23 + ρ2),
FX ∝ Fρ1/2eρ2/2,
(42)
where the F-term order parameter FX is defined by
1
FX = − √ eκ2K/2
2
∂2K
∂ X ∂ X¯
−1 ∂ W¯
∂ X¯
+ κ2 ∂K W¯ . (43)
∂ X¯
Therefore, at the local maximum, FX /D is of order O((ξ +
1)−1/2 F 2/q2) because ρ there is of order O((ξ +1)−1 F 2/q2).
On the other hand, at the global minimum, both D and FX are
of order O(F ), assuming that ρ at the minimum is of order
O(
1
), which is true in our models below. This makes tuning
of the vacuum energy between the F- and D-contribution in
principle possible, along the lines of [1,13].
A comment must be made here on the action in the
presence of non-vanishing F and ξ . 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 ξ contributes to the potential. In this case, the
condition for the local maximum of the scalar potential at ρ = 0
can be satisfied for − 23 < ξ < 0. When F is set to zero, the
scalar potential (27) has a minimum at ρm2in = 23 ln − 23ξ .
In order to have Vmin = 0, we can choose ξ = − 23e .
However, we find that this choice of parameter ξ 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
η-problem in supergravity by taking a linear superpotential,
W ∝ X . In contrast, in the present paper we construct models
with two FI parameters b, ξ in the Kähler frame where the
U (
1
) gauge symmetry is not an R-symmetry. If the new FI
term ξ 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 ξ , 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 η-problem. Nevertheless, we will show below that
this is not the case and the new models with b = 3/2 avoid
the η-problem thanks to the other FI parameter ξ 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 ρ = 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 ξ .
The parameter q controls the overall scale of the potential
and it will be fixed by the amplitude As of the CMB data.
The only free-parameter left over is ξ , which can be tuned to
satisfy the slow-roll condition.
In order to calculate the slow-roll parameters, we need to
work with the canonically normalised field χ defined by
dχ
dρ =
2gX X¯ .
The slow-roll parameters are given in terms of the canonical
field χ by
Since we assume inflation to start near ρ = 0, the slow-roll
parameters for small ρ can be expanded as
η =
4(1 + ξ )
3
+ O(ρ).
Note also that η is negative when ξ < −1. We can therefore
tune the parameter ξ to avoid the η-problem. The observation
is that at ξ = −1, the effective charge of X vanishes and thus
the ρ-dependence in the D-term contribution (27) becomes
of quartic order.
For our present choice F = 0, the potential and the
slowroll parameters become functions of ρ2 and the slow-roll
parameters for small ρ2 read
(46)
η(0)2ρ2 .
(47)
Note that we obtain the same relation between and η as in
the model of inflation from supersymmetry breaking driven
by an F-term from a linear superpotential and b = 1 [1]. Thus,
there is a possibility to have flat plateau near the maximum
that satisfies the slow-roll condition and at the same time a
small cosmological constant at the minimum nearby.
The number of e-folds N during inflation is determined
by
N = κ2
χend
V dχ = κ2
∂χ V
ρend V
∂ρ V
dχ
dρ
χ∗ ρ∗
where we choose | (χend)| = 1. Notice that the slow-roll
parameters for small ρ2 satisfy the simple relation =
η(0)2ρ2 + O(ρ4) by Eq. (47). Therefore, the number of
efolds between ρ = ρ1 and ρ2 (ρ1 < ρ2) takes the following
simple approximate form as in [1],
2
as long as the expansions in (47) are valid in the region ρ1 ≤
ρ ≤ ρ2. Here we also used the approximation η(0) η∗,
which holds in this approximation.
We can compare the theoretical predictions of our model
to the observational data via the power spectrum of scalar
perturbations of the CMB, namely the amplitude As , tilt ns
and the tensor-to-scalar ratio of primordial fluctuations r .
These are written in terms of the slow-roll parameters:
(49)
4
κ V∗ ,
As = 24π 2
∗
ns = 1 + 2η∗ − 6 ∗
r = 16 ∗ ,
(50)
q = 4.544 × 10−7, ξ = −1.005.
(52)
where all parameters are evaluated at the field value at horizon
crossing χ∗. From the relation of the spectral index above,
one should have η∗ −0.02, and thus Eq. (49) gives
approximately the desired number of e-folds when the logarithm is
The scalar potential for this parameter set is plotted in Fig. 2.
The slow-roll parameters during inflation are shown in Fig. 3.
By choosing the initial condition ρ∗ = 0.055 and ρend =
0.403, we obtain the results N = 58, ns = 0.9542, r =
Fig. 3 This plot shows the slow-roll parameters for F = 0, b = 3/2,
ξ = −1.005 and q = 4.544 × 10−7
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∗ 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
r
16(|η∗|ρende−|η∗|N )2
10−4,
H∗
1012 GeV, (51)
where we used |η∗| = 0.02, N 50 − 60 and ρend 0.5,
which are consistent with our models. In the next
subsection, we will present a model which gives a tensor-to-scalar
ratio bigger than the upper bound above, by adding some
perturbative corrections to the Kähler potential.
As an example, let us consider the case where
7.06 × 10−6 and As = 2.2 × 10−9 as shown in Table 1 which
are within the 2σ -region of Planck’15 data as shown in Fig. 4.
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
nonzero cosmological constant at the minimum is to turn on the
superpotential W = κ−3 F = 0, as mentioned in Sect. 3.2.
In this case, the scale of the cosmological constant is of order
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
While the results in the previous example agree with the
current limits on 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 X¯ :
K = κ−2 X X¯ + A(X X¯ )2 + B(X X¯ )3 + b ln X X¯ ,
(53)
while the superpotential and the gauge kinetic function
remain as in Eq. (25). We now assume that inflation is driven
by the D-term, setting the parameter F = 0. In terms of the
field variable ρ, we obtain the scalar potential:
FAig=. 60.5T4h5i,sBpl=ot 0s.h2o3w0s, ξth=es−lo1w.1-r4o0llapnadraqm=ete2r.s12fo1r×F1=0−05, b = 3/2,
V = q2 b + ρ2 +2 Aρ4 +3Bρ6 +ξρ 43b e 23 Aρ4+Bρ6+ρ2 2 .
(54)
We now have two more parameters 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 ρ
in the scalar potential. So, we consider the case b = 3/2.
The simple formula (49) for the number of e-folds for small
ρ2 also holds even when A, B are turned on because the new
parameters appear at order ρ4 and higher. To obtain r ≈ 0.01,
we can choose for example
The scalar potential for these parameters is plotted in Fig. 5.
The slow-roll parameters during inflation are shown in Fig. 6.
By choosing the initial condition ρ∗ = 0.240 and ρend =
0.720, we obtain the results N = 57, ns = 0.9603, r = 0.015
and As = 2.2 × 10−9 as shown in Table 2, which agree with
Planck’15 data as shown in Fig. 7.
Fig. 7 A plot of the predictions for the scalar potential with F = 0,
b = 3/2, A = 0.545, B = 0.230, ξ = −1.140 and q = 2.121 × 10−5
in the ns - r plane, versus Planck’15 results
In summary, in contrast to the model in [1] where the
Fterm 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
ξ ) 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 X¯ (i.e. up to order ρ6). 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.
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.
Open Access This article is distributed under the terms of the Creative
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