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Eur. Phys. J. C (2018) 78:1014

https://doi.org/10.1140/epjc/s10052-018-6485-6

Regular Article - Theoretical Physics

Unitarity-safe models of non-minimal inﬂation in supergravity

Constantinos Pallis

School of Technology, Aristotle University of Thessaloniki, 541 12 Thessaloniki, Greece

Received: 16 October 2018 / Accepted: 28 November 2018 / Published online: 14 December 2018

Abstract We show that models of chaotic inﬂation based

on the φ

potential and a linear non-minimal coupling to

gravity, f

= 1 + c

φ, can be done consistent with data in

the context of Supergravity, retaining the perturbative unitar-

ity up to the Planck scale, if we employ logarithmic Kähler

potentials with prefactors −p(1+n) or −p(n+1)−1, where

−0.035  n  0.007 for p = 2or−0.0145  n  0.006

for p = 4. Focusing, moreover, on a model employing a

gauge non-singlet inﬂaton, we show that a solution to the μ

problem of MSSM and baryogenesis via non-thermal lepto-

genesis can be also accommodated.

1 Introduction

Inﬂation established in the presence of a non-minimal cou-

pling between the inﬂaton φ and the Ricci scalar R is called

collectively non-minimal inﬂation (

nMI)[1–13]. Between the

numerus models, which may be proposed in this context,

universal attractor models (

UAMs)[14] occupy a prominent

position since they exhibit an attractor towards an inﬂation-

ary phase excellently compatible with data [15]forc

 1

and φ ≤ m

– where m

is the reduced Planck mass. UAMs

consider a monomial potential of the type

(φ) = λ

p/2

p−4

(1)

in conjunction with a strong non-minimal coupling [5,14]

(φ) = 1 +c

(φ/m

)

q/2

(2)

with p = q. The emergence of an inﬂationary plateau in these

models can be transparently shown in the Einstein frame (

EF)

where the inﬂationary potential,



attr

, takes the form



attr

= V

 λ

, (3)

e-mail: kpallis@auth.gr

with the exponent in the denominator being related to the

conformal transformation employed [1–5] to move from the

Jordan frame (

JF)toEF.

However, due to the large c

values needed for the estab-

lishment of nMI with φ ≤ m

, the inﬂationary scale,



1/4

attr

turns out to be larger than the Ultraviolet (

UV) cut-off scale

[16–18]



attr

= m

1/(q/2−1)

with 2 < q ≤ 14/3(4)

of the corresponding effective theory, which thereby breaks

down above it. A criticism of these results may be found in

Ref. [19], where background-dependent cut-off scales, well

larger than



1/4

attr

, are evaluated. This practice is rather ques-

tionable, though, preventing the possibility of making per-

turbative extrapolations of the low-energy theory. Indeed,

the low-energy theory expanded around the true low-energy

vacuum should break at a scale that is calculable within the

low-energy ﬁeld expansion [20]. Therefore, the presence of



attr

in Eq. (4) cannot be avoided, and it signals the break-

down of the theory in that ﬁeld range. Several ways have been

proposed to surpass the inconsistency above. E.g., incorpo-

rating new degrees of freedom at 

attr

[21], or assuming

additional interactions [22], or invoking a large inﬂaton vac-

uum expectation value (

v.e.v)





[23–27], or introducing a

sizable kinetic mixing in the inﬂaton sector which dominates

over

[28–34].

Here we propose a novel solution – applied only in the con-

text of Supergravity (

SUGRA) – to the aforementioned prob-

lem, by exclusively considering q = 2inEq.(2) – cf. Ref.

[35]. In this case, the canonically normalized inﬂaton



φ is

related to the initial ﬁeld φ as



φ ∼ c

φ at the vacuum of the

theory, in sharp contrast to what we obtain for q > 2 where



φ  φ. As a consequence, the small-ﬁeld series of the various

terms of the action expressed in terms of



φ, does not contain

in the numerators, preventing thereby the reduction of



below m

[7,18]. The same conclusion may be drawn

within the JF since no dangerous inﬂaton-inﬂaton-graviton

123

1014 Page 2 of 18 Eur. Phys. J. C (2018) 78 :1014

interaction appears because

R,φφ

= 0forq = 2[7]. Note

that the importance of a scalar ﬁeld with a totally or partially

linear non-minimal coupling to gravity in unitarizing Higgs

inﬂation within non-SUSY settings is highlighted in Refs.

[36,37].

A permanently linear

can be reconciled with an inﬂa-

tionary plateau, similar to that obtained in Eq. (3), in the con-

text of SUGRA, by suitably selecting the employed Kähler

potentials. Indeed, this kind of models is realized in SUGRA

using logarithmic or semilogarithmic Kähler potentials [8–

12] with the prefactor (−N) of the logarithms being related

to the exponent of the denominator in Eq. (3). Therefore,

by conveniently adjusting N we can achieve, in principle,

a ﬂat enough EF potential for any p in Eq. (1) but taking

exclusively q = 2inEq.(2). As we show in the following,

this idea works for p ≤ 4inEq.(1) supporting nMI com-

patible with the present data [15]. For p = 4wealsoshow

that the inﬂaton may be identiﬁed with a gauge singlet or

non-singlet ﬁeld. In the latter case, models of non-minimal

Higgs inﬂation are introduced, which may be embedded in a

more complete extension of MSSM offering a solution to the

μ problem [38] and allowing for an explanation of baryon

asymmetry of the universe (

BAU)[39]vianon-thermal lep-

togenesis (

nTL)[40,41]. The resulting models employ one

parameter less than those used in Refs. [33,34] whereas the

gauge-symmetry-breaking scale is constrained to values well

below the MSSM uniﬁcation scale contrary to what happens

in Refs. [27,33,34].

Below, we ﬁrst – in Sect. 2 – describe the SUGRA set-up

of our models and prove that these are unitarity-conserving in

Sect. 3. Then, in Sect. 4, we analyze the inﬂationary dynam-

ics and predictions. In Sect. 5 we concentrate on the case

of nMI driven by a Higgs ﬁeld and propose a possible post-

inﬂationary completion. We conclude in Sect. 6. Unless oth-

erwise stated, we use units where the reduced Planck scale

= 2.4 × 10

GeV is set to be unity.

2 Supergravity framework

In Sect. 2.1 we describe the generic formulation of our models

within SUGRA, and then we apply it for a gauge singlet and

non-singlet inﬂaton in Sects. 2.2 and 2.3 respectively.

2.1 General framework

We focus on the part of the EF action within SUGRA related

to the complex scalars z

– denoted by the same superﬁeld

symbol – which has the form [8–11]

S =





−





−



R + K

g

μν

∗

−





, (5a)

where



R is the EF Ricci scalar curvature, D

is the gauge

covariant derivative, K

= K

∗

, and K

βγ

= δ

– throughout subscript of type , z denotes derivation with

respect to (

w.r.t) the ﬁeld z.Also,



V is the EF SUGRA poten-

tial which can be found once we select a superpotential W in

Eq. (24) and a Kähler potential K via the formula



V = e



∗

− 3|W |





, (5b)

where D

W = W

W is the Kähler covariant deriva-

tive and D

= z

(

)

are the D term corresponding to

a gauge group with generators T

and (uniﬁed) gauge cou-

pling constant g. The remaining terms in the right-hand side

(

r.h.s) of the equation above describes contribution from the

F terms. The contribution from the D terms vanishes for a

gauge singlet inﬂaton and can be eliminated during nMI for

a gauge non-singlet inﬂaton, by identifying it with the radial

part of a conjugate pair of Higgs superﬁelds – see Sect. 2.3.

In both our scenaria, we employ a “stabilizer” ﬁeld S placed

at the origin during nMI. Thanks to this arrangement, the

term 3|W |



V vanishes, avoiding thereby a possible run-

away problem, and the derivation of



V is facilitated since the

non-vanishing terms arise from those proportional to W

and

∗

– see Sects. 2.2.2 and 2.3.2 below.

Deﬁning the frame function as

−/N = exp

(

−K /N

)

⇒ K =−N ln

(

−/N

)

, (6)

where N > 0 is a dimensionless parameter, we can obtain

– after a conformal transformation a long the lines of Refs.

[8–11,29]–theJFformofS which is

S =



√

−g





R + ω

∗

− V

−

A



with ω

= 

3 − N



(7a)

Here we use the shorthand notation 

= 

, and 

¯α



∗¯α

.WealsosetV =



V 

and

=−iN





− 

¯α

∗¯α



/6. (7b)

Although the choice N = 3 ensures canonical kinetic terms

in Eq. (7a), N may be considered in general as a free parame-

ter with interesting consequences not only on the inﬂationary

observables [7,25,26,29,32,42,43] but also on the consis-

tency of the effective theory, as we show below.

123

Eur. Phys. J. C (2018) 78 :1014 Page 3 of 18 1014

2.2 Gauge-singlet inﬂaton

Below, in Sect. 2.2.1, we specify the necessary ingredients

(super- and Kähler potentials) which allow us to imple-

ment our scenario with a gauge-singlet inﬂaton. Then, in

Sect. 2.2.2, we outline the derivation of the inﬂationary poten-

tial.

2.2.1 Set-up

This class of models requires the utilization of two gauge

singlet chiral superﬁelds, i.e., z

= , S, with  (α = 1)

and S (α = 2) being the inﬂaton and a “stabilizer” ﬁeld

respectively. More speciﬁcally, we adopt the superpotential

= λS

p/2

, (8)

which can be uniquely determined if we impose two symme-

tries:

(i) an R symmetry under which S and  have charges

1 and 0;

(ii) a global U(1) symmetry with assigned charges

−1 and 2/p for S and . To obtain a linear non-minimal

coupling of  to gravity, though, we have to violate the latter

symmetry as regards . Indeed, we propose the following

set of Kähler potentials

=−N ln



1 + c

+ F

∗

) − F

−

/N + F



, (9a)

=−N ln



1 + c

+ F

∗

) + F



+ F

−

, (9b)

=−N ln



1 + c

+ F

∗

) − F

−



+ F

, (9c)

=−N ln



1 + c

+ F

∗

)



+ F

−

+ F

, (9d)

=−N ln



1 + c

+ F

∗

)



+ F

. (9e)

Recall that N > 0. From the involved functions

= /

√

2 and F

−

=−



 − 

∗



(10)

the ﬁrst one allows for the introduction of the linear non-

minimal coupling of  to gravity whereas the second one

assures canonical normalization of  without any contribu-

tion to the non-minimal coupling along the inﬂationary path

–cf.Refs.[14,30,31]. On the other hand, the functions F

with l = 1, 2, 3 offer canonical normalization and safe stabi-

lization of S during and after nMI. Their possible forms are

given in Refs. [33,34]. Just for deﬁniteness, we adopt here

only their logarithmic form, i.e.,

=−ln(1 +|S|

/N ), (11a)

= N

ln(1 +|S|

), (11b)

= N

ln(1 + F

−

+|S|

), (11c)

with 0 < N

< 6. Recall [8–11,44–46] that the simplest

term |S|

leads to instabilities for K = K

and K

and light

excitations for K = K

− K

. The heaviness of these modes

is required so that the observed curvature perturbation is gen-

erated wholly by our inﬂaton in accordance with the lack of

any observational hint [39] for large non-Gaussianity in the

cosmic microwave background. Note that all the proposed

K ’s contain up to quadratic terms of the various ﬁelds. Also

(and F

∗

) is exclusively included in the logarithmic part

of the K ’s whereas F

−

may or may not accompany it in

the argument of the logarithm. Note ﬁnally that, although

quadratic nMI is analyzed in Refs. [6,7,14] too, the present

set of K ’s is examined for ﬁrst time.

2.2.2 Inﬂationary potential

Along the inﬂationary track determined by the constraints

S =  − 

∗

= 0, or s =¯s = θ = 0(12)

if we express  and S according to the parametrization

 = φ e

iθ

√

2 and S = (s + i ¯s)/

√

2, (13)

the only surviving term in Eq. (5b)is



V (θ = s =¯s = 0) = e

∗

CI,S

. (14)

which, for the K ’s in Eqs. (9a)–(9e), reads



p/2



for K = K

, K

1forK = K

− K

(15)

where we deﬁne the (inﬂationary) frame function as

=−





Eq.

(12)

= 1 + c

φ. (16)

As expected, f

coincides with

in Eq. (2)forq = 2.

This form of f

assures the preservation of unitarity up to

= 1 as explained in Sect. 1 and veriﬁed in Sect. 3.The

last factor in Eq. (15) originates from the expression of K

∗

for the various K ’s. Indeed, K

along the conﬁguration in

Eq. (12) takes the form





= diag



κ + Nc

/2 f

, K

∗



, (17)

where

∗



1/ f

for K = K

, K

1forK = K

− K

(18)

and

κ =



1/ f

for K = K

, K

1forK = K

, K

(19)

123

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