Generic scalar potentials for inflation in supergravity with a single chiral superfield

Abstract

Abstract

We propose a large class of supergravity models in terms of a single chiral matter superfield, leading to (almost) arbitrary single-field inflationary scalar potentials similar to the F-term in rigid supersymmetry. Those scalar potentials are positively semi-definite (in our approximation), and can preserve supersymmetry at the end of inflation. The only scalar superpartner of inflaton is stabilized by a higher-dimensional term in the Kähler potential. We argue that couplings of the inflaton to other sectors of the particle spectrum do not affect the inflationary dynamics, and briefly discuss reheating of the universe by the inflaton decays.

Published for SISSA by Springer

Received: September 10, 2014

Revised: November 5, 2014 Accepted: November 17, 2014

Published: December 9, 2014

Generic scalar potentials for ination in supergravity with a single chiral supereld

Sergei V. Ketova,b,c and Takahiro Teradad

aDepartment of Physics, Tokyo Metropolitan University, 1-1 Minami-ohsawa, Hachioji, Tokyo 192-0397, Japan

bKavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan

cInstitute of Physics and Technology, Tomsk Polytechnic University,30 Lenin Ave., Tomsk 634050, Russian Federation

dDepartment of Physics, The University of Tokyo,

7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We propose a large class of supergravity models in terms of a single chiral matter supereld, leading to (almost) arbitrary single-eld inationary scalar potentials similar to the F -term in rigid supersymmetry. Those scalar potentials are positively semi-denite (in our approximation), and can preserve supersymmetry at the end of ination. The only scalar superpartner of inaton is stabilized by a higher-dimensional term in the Kahler potential. We argue that couplings of the inaton to other sectors of the particle spectrum do not a ect the inationary dynamics, and briey discuss reheating of the universe by the inaton decays.

Keywords: Supergravity Models, Cosmology of Theories beyond the SM

ArXiv ePrint: 1408.6524

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2014)062

Web End =10.1007/JHEP12(2014)062

JHEP12(2014)062

Contents

1 Introduction 1

2 Designing arbitrary inationary potentials in supergravity 3

3 Conrmation of stabilization 6

4 Impact of matter couplings on inaton dynamics 10

5 Conclusion 12

A Specic stabilization for various Kahler potentials 13

B Supergravity realizations of the deformed Starobinsky models 15

C On the possible origin of the quartic term 16

1 Introduction

Inationary cosmology [15] is now getting established by the recent precise observations of the Universe such as the WMAP [6] and the Planck [7]. For example, the spectral index and the tensor-to-scalar ratio are constrained by the Planck+WP+highL+BAO to ns = 0.9608 [notdef] 0.0054 at 68% CL and r < 0.111 at 95% CL, respectively. On the other

hand, the BICEP2 claimed that they discovered r = 0.16+0.060.05 after foreground subtraction with r = 0 disfavored at 5.9 [8]. The large value of r claimed by the BICEP2 implies a large-eld ination because of the Lyth bound [9]. By the way, the renowned Starobinsky model [1] is fully consistent with the Planck data by predicting a very small tensor-to-scalar ratio r [similarequal] 4 [notdef] 103, being a large-eld inationary model also. There are some arguments

in the literature [1012] that the BICEP2 collaboration underestimated the foreground, so that r [similarequal] 0 may still be consistent with the data. Therefore, the inationary models

with r [lessorsimilar] 0.1 are still alive in the present situation. The actual value of r is going to be established by new observations in a not so distant future.

Under such circumstances, we are interested in embedding the inationary models consistent with current observations into a more general framework motivated by particle physics and a fundamental theory of quantum gravity such as superstrings or M-theory. It is natural to consider supergravity [1315] for that purpose because (i) supergravity emerges as the low-energy e ective action of superstrings, and (ii) the energy scale of ination is higher than the electroweak scale but is lower than the Planck scale where some unknown UV e ects may come in. We pursue minimal realizations of ination in supergravity, by minimizing a number of the matter d.o.f. involved.

1

JHEP12(2014)062

Describing ination and, in particular, a large eld or chaotic ination [16] in super-gravity is known to be non-trivial, because of the presence of the exponential factor and the negatively denite term in the F -type scalar potential. A shift symmetry of the Kahler potential plays the crucial role in the model building of chaotic ination in supergravity [17, 18]. In those pioneering papers yet another chiral supereld (sometimes called the(s)Goldstino or Polonyi supereld) of the R-charge 2, with the vanishing vacuum expectation value, was introduced to allow a positively denite inationary scalar potential. Later on, some extensions of that idea with more general chaotic inationary potentials were introduced in supergravity in refs. [1921]. A di erent approach for ination in supergravity with vector or tensor supermultiplets was proposed in ref. [22], and it was extended to embed arbitrary scalar potentials in refs. [23, 24]. All those methods employ the second supereld, in addition to that containing inaton.

Ination with a single supereld, or sGoldstino ination, was previously studied in refs. [2527], and it was concluded that a large-eld ination is virtually impossible in that case [27] (see also [28]). Recently, new models with a nilpotent chiral Goldstino supereld were proposed [29, 30], which lead to the standard Volkov-Akulov action for Goldstino in nonlinearly realized supersymmetry, and a large eld ination is possible. Though those models have only one dynamical complex scalar, their fermionic sectors are much more complicated. In this paper we adopt the standard (linearly realized) supergravity, with only one chiral (inaton) supereld, eluding the known no-go statements. It is worth mentioning here that there is another minimal scenario in which ination is driven by gravitino condensation [31], though with the use of a dilaton chiral supereld in conformal supergravity, in order to make gravitino lighter than the Planck scale.

In our previous short paper [32] we proposed some new supergravity models, realizing a large eld ination and extending the quadratic [16] and the Starobinsky [1] models, by using a single (inaton) chiral supereld only, in the standard supergravity. The required degrees of freedom, other than those of the standard supergravity including graviton and gravitino, were reduced by half from those available in the literature where either an extra chiral [1721] or an extra vector [2224] supereld are required, in addition to the inaton supermultiplet. A discovery of the fact that a large eld ination is possible in supergravity with a single chiral supereld was exciting, though the scalar potential, which we obtained in a very straightforward way, was not very transparent or illuminating enough, in contrast to the simple scalar potentials of refs. [1921] and refs. [23, 24]. The reason is that the suitable Kahler potentials and superpotentials were found by a trial and error procedure in ref. [32]. Moreover, supersymmetry (SUSY) was broken in the vacuum at the inationary scale in some of the models presented in ref. [32]. Though it is not necessarily a problem, it may be inconsistent with the low-energy SUSY scenario incorporating the gauge coupling unication and reducing the hierarchy problem of scalar masses.

The purpose of this paper is to present a new, special and much larger class of the minimal models, employing a single chiral supereld, which (i) lead to very simple and (almost) arbitrary scalar potentials, and (ii) preserve SUSY at the end of ination. We show that it is possible to get a single eld scalar potential, like the one of global SUSY F -term, in the proposed class of supergravity models when using the stabilization mechanism to be explained in section 3.

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The paper is organized as follows. In the next (and main) section 2 we propose a class of models that lead to nearly arbitrary positively semi-denite inationary potentials, by using only a single chiral supereld. We give further support to our framework by arguing about viability of the stabilization mechanism and stability of ination against possible inaton couplings to other sectors of the theory in sections 3 and 4, respectively. In section 5 we summarize our results. More studies of the viability of our stabilization mechanism for various Kahler potentials are given in appendix A. Some variations of the Starobinsky scalar potential in supergravity with a single chiral supereld are collected in appendix B. In appendix C we propose a mechanism for the possible origin of the key quartic term in the Kahler potential via integration of heavy superelds.

2 Designing arbitrary inationary potentials in supergravity

In ref. [32] our method and results were not practical enough, in order to derive explicit scalar potentials suitable for ination, because the scalar potential derived from our Kahlerand super- potentials of supergravity had a complicated form.

One of the technical reasons was the real part Re that was e ectively xed to a nonzero value 0. It is actually more convenient to redene the supereld so that the vacuum expectation value (VEV) of its leading scalar eld component vanishes. Let us consider the following Kahler potential:1

K = 3 ln

h1 + +

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[parenrightbig]

/p3

[bracketrightBig]

. (2.1)

It has a shift symmetry, ! + ia, with a real parameter a. We implicitly assume

here that there is a stabilization term, like +

4, under the logarithm (it is discussed at length in section 3). The square root factor is introduced to obtain the canonically normalized kinetic terms. The kinetic term and the scalar potential are

Lkin =

1 + +

[parenrightbig]

/p3

2 @ @ , (2.2)

V =

1+ +

[parenrightbig]

/p3

3

0

@[parenleftBig]1+[parenleftBigg] +

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 W

/p3W [vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[parenrightbig]

/p3

p3 1+ +

[parenrightbig]

2 3 [notdef]W [notdef]21

A, (2.3)

respectively. When setting [angbracketleft]Re [angbracketright] = 0, they are simplied to

Lkin = @

@ , (2.4) V = [notdef]W [notdef]2 p3

W W + W W

[parenrightbig]

. (2.5)

Let us now consider the superpotentials having the form

W ( ) = 1

p2

(p2i ), (2.6)

1We take the units where the reduced Planck mass is set to one (MG = MP/p8 = 1) unless it is otherwise stated.

3

where is a real function of its argument [1921] (i.e. all coe cients of its Taylor expansion are real) up to an overall phase that is unphysical. Then the scalar potential is greatly simplied to

V = [notdef]W (iIm )[notdef]2 = [parenleftBig]

2, (2.7)

where the prime denotes di erentiation with respect to its argument, and [notdef] = p2Im is the canonically normalized inaton eld. Demanding a real function may look like a strong condition, but it is obviously satised in the case of a monomial superpotential that is su cient for the simplest chaotic model with a quadratic potential. On the one hand, even if the reality or phase alignment condition is not the case, ination could occur with a straightforwardly obtained but a little more complicated scalar potential (2.5). On the other hand, if it is satised at a high-energy scale, the functional form of is preserved by the non-renormalization theorem [3335].

The superpotential can be viewed as a small explicit breaking of the shift symmetry in the sense of t Hooft [36]. Hence, one expects the shift symmetry breaking terms to be suppressed by the same scale as the superpotential also in the Kahler potential due to quantum corrections. Those e ects are beyond the scope of this paper since we are interested in a simple classical framework in the rst place. See, however, refs. [37, 38] for studies of those extra contributions.

It is also possible to switch the roles of the real and imaginary parts by a eld redenition. When we start with

K = 3 ln

1 + i + i

2, (2.9)

where = p2Re is the canonically normalized inaton eld, and the real function is dened by W ( ) = 1

p2 (p2 ).

This way a very large class of inationary potentials can be obtained by using only one chiral supereld. The only restriction is that the scalar potential should be square of some real function. And it is automatically positively semi-denite that is quite comfortable for phenomenological purposes. It is not di cult to obtain a vacuum with the vanishing cosmological constant also. After that it is always possible to add a constant to the superpotential in order to cancel a SUSY breaking. Adding a constant to the superpotential does not a ect the scalar potential because the latter is determined by the derivative of the former.

Our approach to the inationary model building in supergravity is as powerful as those of refs. [1921, 23, 24] in the sense that the superpotential leading to an arbitrary positively semi-denite scalar potential can be approximately reconstructed by taking its square root, Taylor expanding it, and then integrating. At the same time, our method is more economical in the sense that only a single chiral supereld (other than the standard gravitational multiplet containing graviton and gravitino) is used.

4

[prime] ()

[prime]([notdef])

[parenrightbig]

(2.8)

and a superpotential W with real coe cients, after stabilization of the imaginary part

hIm [angbracketright] = 0, the scalar potential reads

V = [notdef]W (Re )[notdef]2 = [parenleftBig]

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/p3

[parenrightBig]

Example 1: a monomial potential. The 2n-th power monomial potential V =

|c2n[notdef]22n follows from the following superpotential (in the notation of eqs. (2.8) and (2.9)):

W = 2n/2c2nn + 1 n+1. (2.10)

In particular, a quadratic potential V = m22/2 is obtained from the superpotential

W = 12m 2. (2.11)

Example 2: the Starobinsky potential. The Starobinsky inationary scalar potential is reproduced by the superpotential

W = p3

2 m +

p3

2 e2 /p3 1[parenrightBig][parenrightBigg]

. (2.12)

When using the current framework, its easy to obtain a set of the deformed Starobinsky models [23, 39] also. The -deformed superpotential

W = p3

2 m +

p3

2 e2 /p3 1[parenrightBig][parenrightBigg]

(2.13)

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leads to the scalar potential

V = 3

4 m2

1 e

p2/3

2. (2.14)

In appendix B we demonstrate some other ways of getting the similar (or the same) deformed Starobinsky potentials generalizing those of ref. [32].

Example 3: a symmetry breaking potential. The symmetry breaking-type potential (our inaton is assumed to be a singlet),

V = 2 v2

2 , (2.15)

can be used for a new [4, 5], chaotic [16], or topological [4042] ination, depending on the values of the parameters and the initial conditions, see e.g. a review [43] . It is obtained from the following superpotential:

W = p

23 3 v2 [parenrightbigg]

. (2.16)

Example 4: a sinusoidal potential. The sine-modulated inationary scalar potential V = V02 (1 cos n) for natural ination [44, 45] follows from the superpotential

W = p2V0

n

q1 cos p2n cotn

p2 . (2.17)

5

3 Conrmation of stabilization

To demonstrate viable stabilization, we assume a specic Kahler potential and a generic superpotential,

K = 3 ln 1 +

+

+ +

4p3 [parenrightBigg]

, (3.1)

W = 1

p2

(p2i ). (3.2)

Note that the stabilization term (proportional to ) does not break the shift symmetry.

Other symmetry-preserving terms, ( +

)n, may appear in the Kahler potential. In the presence of such terms, ination can still be realized as long as the non-inaton eld is stabilized, but the resulting inaton potential will be corrected and become complicated. This type of the stabilization term was rst introduced in ref. [46] and recently was used in refs. [32, 39, 47, 48]. Though some explanations of the mechanism are given in those references,2 we analyze it here again, in our specic setup, for the sake of self-completeness and transparency. The Kahler metric is

K

=

1 12p3 +

2 4

+

3 + 4 2

+

6

JHEP12(2014)062

2 . (3.3)

We assume [greaterorsimilar] O(1) so that the real part (non-inaton) must be smaller than one, in

order to keep the canonical sign of the kinetic term. The scalar potential is

V = A1

hB1 [notdef]W [notdef]2 p3 [parenleftBig]1 + 8p2 3[parenrightBig]

B2 W W + W

W

1 + + + ( + )4 p3

[parenrightbig]

v +72 2

p3 + p2 + 4 4[parenrightBig]

B3 [notdef]W [notdef]2[bracketrightBig]

, (3.4)

where = 1

p2 ( + i[notdef]) with and [notdef] real, and

A = 1 24p3 2 8p2 3 + 32 26 , (3.5)

B = 1 + p2 + 4 4

p3 . (3.6)

The expectation value of is obtained by truncating the higher order terms beyond

O() in the stationary condition V = 0,

= 4p6

[prime]([notdef])2 3p6

([notdef]) [prime][prime]([notdef])

2

108p3 ([notdef])2+ 72p3 +14

[parenrightbig]

[prime]([notdef])212

([notdef]) [prime][prime]([notdef])+3 [prime][prime]([notdef])23

[prime]([notdef]) [prime][prime][prime]([notdef])

[parenrightBig]

2As was pointed out to us by the referee, our setup is di erent from the case of two superelds, where the quartic term of the stabilizer (Polonyi) eld appears in the low-energy e ective theory of the ORaifeartaigh model coupled to supergravity [49]. In appendix C we briey discuss the possible origin of the shift-symmetric quartic term of used in our approach.

6

4p6 3p3[epsilon1]E

[similarequal] 4 54p3 E + 36p3 3p2[epsilon1]E + 7[parenrightBig]

, (3.7)

where [epsilon1] = 12(V [prime]([notdef])/V ([notdef]))2 is the slow-roll parameter. The rst equality holds both during ination and at the vacuum. In the second equality, we have used V [similarequal] [notdef]

[prime]([notdef])[notdef]2 and have neglected the terms proportional to the slow-roll parameters, unless they are accompanied by the enhancement factor

E

([notdef])

[prime]([notdef])

!2, (3.8)

JHEP12(2014)062

so that it is valid during ination. For example, in the monomial superpotential case, E = ~

n

2 and it is large (E > 1) during the large eld ination ([notdef][notdef][notdef] > 1) for n of the order one. In the case of the Starobinsky potential (2.12), we nd E =

[parenleftBig]

~

2, and

it is also large during ination. Thus, typically, E is of the order [notdef]2, and p[epsilon1]E is roughly of the order one. In summary, we have

[similarequal] O 102 1E1 [parenrightbig]

O 102 1[notdef]2

1e

p2/3~

p3 2

[parenrightbig]

, (3.9)

during ination. It is smaller than one indeed, being also consistent with the truncation above. The kinetic term is approximately canonical.

The mass squared of is

V = 13 216p3 E + 144p3 + 28 12p2[epsilon1]E + 6[epsilon1] 3 + O( E)[parenrightBig]

[prime]([notdef])2

[similarequal] 2

108p3E + 72p3[parenrightBig]

H2, (3.10)

where = V [prime][prime]([notdef])/V ([notdef]) is another slow-roll parameter. The O ( E) term is of the order

one, and is neglected in the last expression together with other subdominant terms. The mass of can be easily larger than the Hubble scale H. In this way the real part (noninaton) can be stabilized.

The smallness itself of compared to one does not ensure validity of approximation in the previous section because the small nonzero value may break cancellation among terms in the scalar potential due to the no-scale structure. Now we check that the corrections to the scalar potential (2.7) induced by the nonzero value of in eq. (3.7) are actually smaller than the leading terms. The scalar potential (up to the leading corrections) is given by

V = [prime] ([notdef])2 [parenleftBig]

4 [prime] ([notdef])2 3

([notdef]) [prime][prime] ([notdef])

2

216p3 ([notdef])2+ 144p3 +28

[parenrightbig]

[prime] ([notdef])224

([notdef]) [prime][prime] ([notdef])+6 [prime][prime] ([notdef])26

[prime] ([notdef]) [prime][prime][prime] ([notdef])

8 3p2[epsilon1]E[parenrightBig]2

[similarequal]

[prime] ([notdef])2

1

. (3.11)

0 16 54p3E + 36p3 3p2[epsilon1]E + 7[parenrightBig]

1

C

A

The rst equality holds both during ination and at the vacuum, whereas the second equality is based on the same approximation in the second equality of eq. (3.7) (valid during large eld ination). The corrections are indeed subdominant and vanish in the limit of large E (large [notdef]) or large with xed [epsilon1]E. For example, the numerical value of

7

JHEP12(2014)062

Figure 1. The scalar potential of the stabilised quadratic model. The mass scale and the stabilization strength are set to m = 105 and = 1, respectively.

the second term in the parenthesis of eq. (3.11) is 2.93 [notdef] 103 for (E = 5, [epsilon1] = 0.1,

and = 1), and 8.61 [notdef] 103 for (E = 10, [epsilon1] = 0.1, and = 0.1). These arguments

justify our treatment of the theory as the e ective single eld inationary theory where the kinetic term is approximately canonically normalized and the scalar potential is given by V [similarequal] [notdef]

[prime]([notdef])[notdef]2 in the large eld regime of the inaton [notdef].

To convince a reader even more, we calculate numerically the dynamics of ination. We take two benchmark models as the examples: (i) the chaotic ination with a quadratic potential, and (ii) the Starobinsky potential. We set the inaton mass and the stabilization parameter as m = 105 and = 1 for simplicity. Note that the stabilization parameter of the order one works pretty well, as is shown below.

Let us consider the example (i): the chaotic model with a quadratic potential. The potential of the model with the stabilization proposed in this section is shown in gure 1. The trajectory of the inaton eld in this model is shown in gure 2. We apply the initial condition away from the stabilization valley in order to check how the stabilization mechanism works. The real part (non-inaton) rapidly oscillates around (damped to) the instantaneous minimum, and after that the trajectory is approximately that of single-eld ination. It slightly deviates from the imaginary axis near the end of ination, and nally oscillates around the vacuum. The deviation is smaller in the larger inaton eld value (gure 2) because of eq. (3.9). The fractional di erence between the inaton scalar potential along the trajectory and the quadratic potential is only within 1.4% or even smaller well before the end of ination.

Next, let us consider the example (ii): the Starobinsky model in our framework. The potential of the model with the stabilization of this section is shown in gure 3. The trajectory of the inaton eld in the model is shown in gure 4. It is qualitatively similar

8

Figure 2. The inaton trajectory (green) in the stabilized quadratic model. The initial conditions are = 0.14, [notdef] = 15,

= 0, and

[notdef] = 0. The mass scale and the stabilization strength are set to m = 105 and = 1, respectively. The contour plot of logarithm of the potential is shown in purple.

Figure 3. The scalar potential of the stabilized Starobinsky model. The mass scale and stabilization strength are set to m = 105 and = 1, respectively.

to the case of the quadratic model (see gure 2). The fractional di erence between the inaton scalar potential along the trajectory and the Starobinsky potential is only within 2.2%, or even smaller well before the end of ination.

9

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Figure 4. The inaton trajectory (green) in the stabilized Starobinsky model. The initial conditions are = 0.14, [notdef] = 5.7,

= 0, and

[notdef] = 0. The mass scale and the stabilization strength are set to m = 105 and = 1, respectively. The contour plot of logarithm of the potential is shown in purple.

4 Impact of matter couplings on inaton dynamics

We realized ination in supergravity with a single chiral supereld. After all we must couple the inaton sector to other matter such as the Standard Model sector or a hidden sector where SUSY is broken. In this section we consider a few simple ways of coupling and check whether they a ect the inaton dynamics. We also discuss the inaton decay briey.

We assume that the inaton superpotential and a superpotential of other superelds are decoupled,

W ( , i) = W (inf)( ) + W (other)(i), (4.1)

where is the inaton and i stand for the particle species i other than inaton. This form is preserved during the renormalization group running due to the non-renormalization theorem [3335].

First, let us consider the case when the Kahler potential of inaton and that of the other elds are also decoupled,

K( , i,

,

j) = 3 ln [parenleftbigg]

1 + 1

p3 +

JHEP12(2014)062

[parenrightbig][parenrightbigg]

j), (4.2)

where we have implicitly assumed the existence of a stabilization term under the logarithm. This form is not preserved under the renormalization group running, but we take it just as a simple example here. Then the derivatives with respect to both the inaton and the other elds vanish, so that there is no kinetic mixing between the inaton and the other

10

+ K(other)(i,

elds. The scalar potential with the ( +

V = eK(other)

d2 2E c WAWA + h.c. , (4.4)

where c is the coupling constant and WA is the supereld strength of a real supereld.

Although this coupling breaks the shift symmetry,3 it could be generated with real c as the anomaly of an underlying symmetry in the UV theory [53]. The decay rate is Ng[notdef]c[notdef]2m3~/128M2G [54, 55], where Ng is the number of generators of the gauge algebra,

m~ is the inaton mass, and MG is the reduced Planck mass. Note that the inaton cannot decay through the super-Weyl-Kahler and sigma-model anomaly e ects [5658], because the Kahler potential does not depend on the inaton. The two-body decay rates into scalars and spinors are of order m~m2/M2G [57, 58], where m is the mass of daughter particles. The three-body decay rate is sizable of the order [notdef]yt[notdef]2m3~/M2G [57, 58],

where yt is the top Yukawa coupling. The decay rate into a pair of gravitinos is

[vextendsingle][vextendsingle][vextendsingle]

G(e )~[vextendsingle][vextendsingle][vextendsingle]

2 m5~/288m23/2M2G [54, 55] with the e ective coupling constant in our model be

ing evaluated as

[vextendsingle][vextendsingle][vextendsingle]

G(e )~[vextendsingle][vextendsingle][vextendsingle]

2[52, 59, 60], where m3/2 is the gravitino mass, and mz is the mass of the SUSY breaking eld z.

3If the coe cient c is real (in our convention transforms in the imaginary direction under the shift symmetry), the shift symmetry is broken only via non-perturbative e ects.

m3/2 m~

) being stabilized at the origin is

[notdef]W [notdef]2 p3 [parenleftBig]

W (other) W + W (other)W

[parenrightBig]+ K(other)jiDiW Dj W

[parenrightBig]

, (4.3)

where DiW = Wi +KiW is the covariant derivative. There are the Hubble induced masses p3H for the scalars other than the inaton, so light elds are frozen at the origin during ination, whereas heavy elds are decoupled because of their own masses. Therefore, the dynamics is essentially the single-eld ination. If the SUSY breaking scale W (other) is high,

the inaton potential receives corrections. This feature is similar to the models with several chiral superelds see the discussions of SUSY breaking on ination in these models in refs. [50, 51]. The impact of the conformal rescaling, required to move to the Einstein frame, on SUSY breaking term is very well controlled in the sense that the conformal factor does not depend on the inaton because of the shift symmetry [51]. Corrections proportional to the inaton superpotential also arise from the third term in eq. (4.3) in the case there is a large Ki (large VEV in the case of minimal Kahler potential). This is in contrast to models with the sGoldstino [angbracketleft]S[angbracketright] = 0 and the superpotential W / S because the

value of the superpotential vanishes in these models. In summary, the inaton potential is not a ected by matter coupling (4.2) if the SUSY breaking scale is low and there is no large VEV. The latter condition is satised due to the Hubble-induced mass. Further quantitative study will be done elsewhere. Note that high-scale SUSY breaking (inaton mass less than the mass of the SUSY breaking eld) is also disfavored from the perspective of gravitino overproduction from inaton decay see the text below and ref. [52] for more.

Inaton can decay into gauge bosons and gauginos, if there is a coupling like

1 4

[integraldisplay]

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2 = 27

2[parenleftBig]m2z m2~m 2z

11

Next, let us consider the case when the Kahler potentials are summed under the logarithm,

K( , i,

,

j) = 3 ln [parenleftbigg]

1 + 1

p3 +

, (4.5)

where J is a hermitian function. This structure can be understood e.g., as the geometrical sequestering of the inaton sector and the other sectors [61]. Again we have implicitly assumed the presence of a stabilization term. For simplicity of our notation, we introduce a function such that K = 3 ln or = exp(K/3). The Kahler metric and its inverse are

KI J = 2

1 1p3 J

j

[parenrightbig]

1 3J(i,

1p3 Ji Ji

j + 13JiJj

!, K JI =

+ 13JiJi1 p3 Ji 1p3 Jj Jji

!, (4.6)

where capital Latin indices I, J, . . . run over and i, j, . . . , the Jji is the inverse matrix of Jj, while the indices are raised and lowered by those matrices, e.g., Ji = JjiJj. Then the scalar potential is given by

V = 2

[parenleftbigg][parenleftbigg]

+ 13JiJi[parenrightbigg] [notdef]

W [notdef]2 p3 [parenleftBig]

W (other) W

+ W (other)W

[parenrightBig]

+ 1

p3 [parenleftBig]

JiWi W

+ J WiW

[parenrightBig]

+ JjiWi Wj

. (4.7)

The Hubble induced mass is p2H, so the elds other than the inaton are stabilized at their origin during ination. Assuming that these elds are charged under some unbroken symmetries, the rst derivatives Ji vanish. Then the kinetic mixing e ects become negligible. Similar comments to the case of minimal coupling (4.2) apply here too, but there are no terms proportional to the inaton superpotential in eq. (4.7) in the case of sequestered coupling (4.5) (we have again used the phase alignment condition for the inaton superpotential). Inaton dynamics is not a ected by matter coupling if the SUSY breaking scale is low.

The inaton decay is similar to the previous example, but there are no sizable three-body decays (see ref. [62] for a similar situation). The two-body decay rate into scalar particles is of the order [notdef]Jij[notdef]2m3~/M2G. The e ective coupling constant for decay into two gravitinos is

[vextendsingle][vextendsingle][vextendsingle]

G(e )~[vextendsingle][vextendsingle][vextendsingle]

JHEP12(2014)062

[vextendsingle][vextendsingle][vextendsingle]

2 = Jz + 2Wzm~[parenrightBig]m2z m2~m 2z

[vextendsingle][vextendsingle][vextendsingle]

2

.

5 Conclusion

In this paper we proposed the simple framework for a construction of arbitrary positively semi-denite single eld inationary potentials in supergravity by using only a single chiral supereld. The scalar potential see eqs. (2.7) and (2.9) in our framework has (approximately) the same form as the F -term in global SUSY theory, and it e ectively becomes a function of a single eld (inaton) due to the stabilization mechanism. The inaton does not break SUSY at the vacuum. We veried that our stabilization works, and we also proposed some simple ways of coupling the inaton sector to other matter sectors, without a ecting the inationary dynamics.

Our class of the very economical models provides vast possibilities for realizing cosmological ination in supergravity, which are consistent with the observational data in the most minimal setup (with a single chiral supereld).

12

Acknowledgments

SVK was supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, by the World Premier International Research Centre Initiative (WPI Initiative), MEXT, Japan, and by the Competitiveness Enhancement Program of the Tomsk Polytechnic University in Russia. TT is grateful to colleagues at the University of Tokyo, especially to Koichi Hamaguchi and Kazunori Nakayama, and also to Shuntaro Aoki and Yusuke Yamada, for useful discussions. TT was supported by a Grant-in-Aid for JSPS Fellows, and a Grant-in-Aid of the JSPS under No. 2610619.

A Specic stabilization for various Kahler potentials

In this paper we focused on the special Kahler potential and a class of the superpotentials such that an arbitrary scalar potential is readily available, but there are still many possibilities leading to a single eld ination from supergravity with a single chiral supereld, as was already explained in ref. [32]. Here we examine the stabilization quality in those theories, and justify our treatment.

Let us begin with the minimal Kahler potential (see eq. (9) in ref. [32]),

K = 12 +

JHEP12(2014)062

4 . (A.1)

The inaton is Im , and Re is stabilized by the term. The scalar potential for a general superpotential W ( ) reads

V = e2 2

2

+ 2 0

0 (2Re 2 0)4

1 12 (2Re 2 0)2 [parenleftBig][notdef]

W [notdef]2 + 2 [parenleftBig]

Re 2 (2Re 2 0)3[parenrightBig][parenleftBigg][parenleftBigg]W

W + W

W

[parenrightbig]

+4 Re 2 (2Re 2 0)3[parenrightBig]2 [notdef]

W [notdef]2[parenrightbigg] 3e2 20 (2Re 2 0)4[notdef]W [notdef]2 . (A.2)

If a deviation of [angbracketleft]Re [angbracketright] from 0 is small, it merely results in a small change of the

coe cient at each term in the above expression, so that the inaton dynamics receives only a minor change. In fact, the expectation value of Re around 0 is found to be small indeed, similarly to the analysis in section 3,

hRe [angbracketright] 0 [similarequal]

0 4 20 1

96 20 + 16 40 + 8 20 1 [similarequal] O[parenleftBigg] 101 1

(A.3)

for 0 [similarequal] O(1), where we have neglected the subdominant powers in Im (inaton) on

dimensional reasons, [notdef]W [notdef] [notdef]W/ [notdef], [notdef]W [notdef] [notdef]W/ 2[notdef], etc. In this case, a relatively large

is required to suppress the deviation (A.3), e.g. [similarequal] O(10) for [angbracketleft]Re [angbracketright] 0 [similarequal] O(102). It is

worth noticing that a correction to the kinetic term, 12 (2Re 2 0)2, is also suppressed

to be O(101 1). The mass squared of the non-inaton is

V [similarequal]

6 16 40 + (96 + 8) 20 1

2 4 0 3

H2, (A.4)

13

where = ( + i[notdef])/p2, and we have used the slow-roll Friedmann equation, V [similarequal] 3H2, and V [similarequal] e2 2

0 (4 20 3)[notdef]W [notdef]2. This mass can be larger than the Hubble scale with moderate

values of 0 and .Next, let us consider the logarithmic Kahler potential (see eq. (25) of ref. [32]),

K = 3 ln

" + + +

2 0

[bracketrightBigg]

4

. (A.5)

The inaton is Im , and Re is stabilized by the term. This is related to eq. (3.1) via eld redenition, but we do not make assumptions about a superpotential here. The scalar potential is (see eq. (26) in ref. [32])

V = 9

+ + ( + 2 0)4

2

3

JHEP12(2014)062

1

[notdef] 1 4 ( +

2 0)3 + 4 2( +

2 0)6 24 0 +

2 0

2

[notdef]

h( + + ( + 2 0)4) [notdef]W [notdef]2 3 1 + 4 ( +

)3

[parenrightbig][parenleftBigg][parenleftBigg]

W W + W W

2 0)2[notdef]W [notdef]2[bracketrightBig]

[parenrightbig]

+ 108 ( +

. (A.6)

After the canonical normalization of the real part, Re = 0ep2/3, the deviation is evaluated as

h[angbracketright] [similarequal]

p6 W W

+ W W

2 288 0 [notdef]W [notdef]2 O(102 1Im 1). (A.7)

This gives rise to the term proportional to [notdef]W [notdef]2 which is absent in the ideal case Re = 0,

and is actually subdominant here. The mass squared of the non-inaton is

V [similarequal] 1296 [notdef]W [notdef]2, (A.8) so that there is no di culty to make the mass larger than the Hubble scale.

Finally, let us consider the Kahler potential used in section 3 of ref. [32],

K = 3 ln

" + + i + i

[bracketrightBigg]

2 0

4

. (A.9)

In this case the imaginary part Im is stabilized and the real part Re is used as the inaton. The scalar potential is

V = 9

+ + (i + i 2 0)4

2

3

1

[notdef] 1 12 +

(i + i

2 0)2 + 4 2 i + i

2 0

6

[notdef]

h( + + (i + i 2 0)4) [notdef]W [notdef]2 3

W W + W W

[parenrightbig]

+12i (i + i

2 0)3 W

W

W W

[parenrightbig]

+ 108 (i + i

2 0)2[notdef]W [notdef]2

[bracketrightbig]

. (A.10)

14

The deviation of Im is obtained as

hIm [angbracketright] 0 [similarequal] i

3 W W W

W

+ 2Re W

W W

W

864 [notdef]W [notdef]2 576 Re

W W + W W

+ 384(Re )2 [notdef]W [notdef]2

iO(102 1(Re )2). (A.11) This ensures that the correction terms are subdominant again. The mass squared of the imaginary part [notdef] =

p2/32 20Im is

V~~ [similarequal] 3 864[notdef]W [notdef]2 576Re

W W + W W

[parenrightbig]

+ 384 (Re )2 [notdef]W [notdef]2[parenrightBig]

, (A.12)

and appears to be O(102 (Re )3) times larger than the Hubble scale squared.

According to this appendix and section 3, the non-inaton eld can be strongly stabilized with the parameter that is not much larger than one, which justies our basic demand for the inationary supergravity model to be treated as a single-eld ination.

B Supergravity realizations of the deformed Starobinsky models

In this appendix we demonstrate the other two ways (i.e. di erent from that in section 2) to obtain the deformed Starobinsky potentials generalizing the models described in ref. [32].

First, let us recall eq. (32) of that paper, where we employed the no-scale Kahler potential and the superpotential containing a term with a negative power n = 1. Let

us now generalize it to an arbitrary negative power as follows:

K = 3 ln

[bracketleftbig][parenleftBigg]

+

JHEP12(2014)062

[parenrightbig]

/3

[bracketrightbig]

, (B.1)

W = 1

ncn n + c0 +

13c3 3 . (B.2)

After stabilization of the imaginary part, the scalar potential becomes

V = a + ben + ce(n+3) + de(2n+3) , (B.3)

where =

p3/2 ln Re is the canonically normalized inaton, and a = 27Re(c0c3)/2, b = 9(1+3/n)Re(cnc3)/2, c = 27Re(c0cn)/2, and d = 9(1+3/n)[notdef]cn[notdef]2/2. The potential

is shown in gure 5. It is possible to choose a = d = b = c > 0, which ensures V = 0

in the vacuum = 0.

Finally, let us vary the parameter a of eq. (34) in ref. [32]. The Kahler potential is taken to be the minimal one with the shift symmetry. We redene the normalization of a here and take

W = m

hb eip2a( 0)

[bracketrightBig]

. (B.4)

The scalar potential for [notdef] = p2Im is then given by

|m[notdef]2e2 20 V = 4 20 3

[parenrightbig][parenleftBigg][parenleftBigg]Reb

ea~

2 0Imb p2aea~[parenrightBig]2 3(Imb)2. (B.5)

There exist solutions for Reb and Imb that lead to the scalar potential

V = e2 20 [notdef]m[notdef]2 4 20 3 + 2a2

[parenrightbig][parenleftBigg][parenleftBigg]1

ea~

2 +

2 . (B.6)

The shape of this potential is displayed in gure 6.

15

JHEP12(2014)062

Figure 5. The deformed Starobinsky potential (B.3) derived from the superpotential containing a negative power (B.2). The powers are 0.03, 0.1, 0.3, 1, 3, and 10 from the bottom to the

top. The parameter values are taken as a = b = c = d = 1.

Figure 6. The deformed Starobinsky potential (B.6) derived from the superpotential (B.4). The parameter a is set to 0.1, 0.3,

p2/3 (Starobinsky), 3, and 10 from bottom to top. The height of the potential is normalised to one.

C On the possible origin of the quartic term

The quartic term in the Kahler potential plays the key role in stabilization of the noninaton scalar of the inaton supereld in our class of models. The simplest interpretation of the quartic term may be by assuming its presence at the tree level. The quartic term respects the shift symmetry, while all kinds of terms allowed by the symmetries of the theory should generically be included into the e ective eld theory. The remaining questions are (i) how much the quadratic and cubic terms have to be suppressed, and (ii) what is the mechanism for their suppression. It requires a separate investigation. However, as a preliminary test, we nd that small quadratic and/or cubic terms destroy the cancellation of the no-scale type model. Hence, our Kahler potential should be regarded as a tuned one, in order to suppress the quadratic and cubic terms. Without such tuning, the theory describes more general inationary models in supergravity [32].

The quartic term may also originate from some UV-completion of our phenomenological supergravity description, such as superstring theory. Unfortunately, exact superstring

16

corrections are not available in the literature. Though a thorough study of the origin of the quartic term from superstring theory is beyond the scope of this paper, we briey discuss the possible origin of the quartic term in the QFT framework, which serves as an existence proof.

In the following, we consider coupling of the inaton to other superelds, and discuss the e ective terms to be obtained through integrating out these superelds. The idea is to give the ( +

)-dependent masses to the other superelds. The quantum correction to the frame function in Jordan frame is known, whose expression depends on the masses of the elds in the Jordan frame. After integrating out these elds, the quantum corrected frame function = 3 exp K +

JHEP12(2014)062

[parenrightbig]

/3

[parenrightbig]

is left. See appendix B of ref. [63] for the case of the quartic stabilization term for the stabilizer eld.

The quartic term must preserve shift symmetry, so we cannot introduce a coupling to the superpotential that is holomorphic.4 Therefore, we consider a coupling in the Kahler potential. Let us suppose the following Kahler potential:

K = 3 ln ( /3) = 3 ln

1 + 1p3 +

[parenrightbig]

1 3A( +

)J(X, X)

, (C.1)

where A = A( +

) is a function of +

, and J is the kinetic function for other superelds. If breaks SUSY, it gives X mass by the term like A[prime][prime][notdef]F [notdef]2[notdef]X[notdef]2, but we do not want SUSY to be broken above the ination scale.

Let us take a simple superpotential for X, W = 12mX2, where m is a mass parameter much larger than the inaton mass. Masses squared divided by the coe cient of the kinetic terms for scalar and spinor particles are approximately m20 = 2m2/9A and m21/2 = 3m2/27A2, respectively, in the Jordan frame. Then the one-loop correction to the frame function is [63]

=

1162 (1 A) m20 ln [parenleftbigg]

m20

[notdef]2

[parenrightbigg]

+ m21/2 ln

m21/2

[notdef]2

[parenrightBigg][parenrightBigg]

. (C.2)

After expanding A as A +

= 1 + c1 +

+ c2 +

2 + . . . , one can easily see that the quartic term appears, as well as the higher and lower order terms. The full expression is long and is not illuminating. The higher order terms do not a ect our inationary dynamics as far as it is expanded around the VEV. The zeroth and rst order terms correspond to renormalization of the Newton constant and the eld. The coe cients of quadratic and cubic terms can be eliminated at some renormalization scale [notdef] by tuning c2 and c3 in the non-minimal coupling function A +

.

Hence, as was anticipated above, it is possible to obtain the quartic term in +

from quantum corrections of heavy elds, with some tuning to suppress the unwanted terms. The origin of the non-minimal coupling in eq. (C.1) should be sought in an UV-complete framework.

4An exception could be the form of W e W0(X) (shift of changes the phase of the superpotential;

X collectively denotes other superelds). However, can be moved into the Kahler potential by a Kahler transformation, K K

original + +

and W W0(X).

17

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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