ACT-7-20

MI-TH-2030

Cosmology of the string derived flipped $SU(5)$

I. Antoniadis<EMAIL_ADDRESS>D.V. Nanopoulos
<EMAIL_ADDRESS>J. Rizos<EMAIL_ADDRESS>
a Laboratoire de Physique Théorique et Hautes Energies - LPTHE
Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France;
Albert Einstein Center, Institute for Theoretical Physics, University of Bern,
Sidlerstrasse 5, 3012 Bern, Switzerland
b George P. and Cynthia W. Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA;
Astroparticle Physics Group, Houston Advanced Research Center (HARC)
Mitchell Campus, Woodlands, TX 77381, USA;
Academy of Athens, Division of Natural Sciences, Athens 10679, Greece
c Physics Department, University of Ioannina, 45110, Ioannina, Greece

We study the cosmology of a string derived supersymmetric flipped $SU(5)$
model in the context of free-fermionic heterotic constructions that allow full
calculability of the effective supergravity in perturbation theory around the
fermionic vacuum where all string moduli have fixed values. The model has 3
generations of chiral families and a Higgs sector leading to particle
phenomenology consistent with low energy data, that has been extensively
studied in the past. Here, we show that it can also accommodate a novel
successful cosmology, based on the no-scale effective supergravity derived
from string theory as well as an appropriate induced superpotential suppressed
by five powers of the string scale. It utilises two gauge singlet chiral
superfields present in the low energy spectrum: the inflaton $y$, identified
as the superpartner of a state mixed with R-handed neutrinos, and the
goldstino $z$ with a superpotential of the form $W_{I}=M_{I}z(y-\lambda
y^{2})$ (in supergravity units) where $\lambda$ is a dimensionless ${\cal
O}\left(1\right)$ parameter and $M_{I}$ the mass scale of inflation generated
at 5th order by the breaking of an anomalous $U(1)_{A}$ gauge symmetry,
characteristic of heterotic string chiral vacua. The resulting scalar
potential leads to Starobinsky type inflation. Our results can be easily
generalised to a large class of models with similar properties.

###### Contents

  1. 1 Introduction
  2. 2 The “revamped” flipped $SU(5)\times U(1)$ string model
  3. 3 Starobinsky type inflation from no-scale supergravity
  4. 4 Realisation of a Starobinsky-like inflationary scenario in the flipped ${SU(5)}\times{U(1)}$ string model
  5. 5 Conclusions and outook
  6. A String construction and spectrum of the flipped $SU(5)\times{U(1)}$ model
  7. B Calculation of the potential in no-scale supergravity

##  1 Introduction

If string theory describes nature unifying all fundamental interactions, it
should be able to explain all physical phenomena at all scales. During the
last three decades, string phenomenology made several steps towards this goal
in parallel with the progress made in understanding the foundations of the
theory. It started with string constructions of particle physics models to
continue with concrete models of cosmology. Very little has been done though
in explicit constructions of both at the same time. Usually, models of
particle physics focus on spectra extending the Standard Model content and
interactions to higher energies, while models of early cosmology focus on
generating the inflaton potential following moduli stabilisation, ignoring the
details of the actual unification of all interactions.

In this work, we make a step towards a simultaneous description of particle
phenomenology and cosmology using the same string derived model[1] within the
framework of free-fermionic formulation of four-dimensional (4d) heterotic
superstrings[2]. Although one of the earliest formalisms of realistic string
model building, it has the advantage of full calculability in perturbation
theory around a vacuum where all moduli except the dilaton are fixed at the
fermionic point, characterised by a set of gauge and discrete symmetries.
Instead of constructing a new model from scratch, we consider one of the very
first successful constructions of the flipped $SU(5)\times U(1)$ [1], that was
already shown to give a remarkable good particle phenomenology [3, 4, 5, 6]
and we study its cosmology. Our analysis can also serve as a proof of concept
for a class of 4d string models with similar properties.

Our goal is to identify the inflaton among the gauge singlet massless spectrum
of the model and derive an inflationary potential consistent with present
observations. We focus on Starobinsky-type inflation [7], which was shown [8,
9, 10, 11, 12, 13] to be naturally embeddable in the framework of $N=1$ no
scale supergravity [14] which is the generic low-energy effective field theory
of string models and in particular of fermionic constructions where it is
calculable to all orders in the string slope parameter $\alpha^{\prime}$ [15,
16, 17].

Following the proposal of Ref. [9, 18, 19, 20], we identify the inflaton $y$
with the superpartner of a state mixed with Right-handed neutrinos and look
for generating a superpotential of the form $W_{I}=M_{I}z(y-\lambda y^{2})$,
in supergravity units, where $z$ is the goldstino superfield, $M_{I}$ the
inflation scale and $\lambda$ a dimensionless tuneable parameter of order
${\cal O}\left(1\right)$. Since the scale of inflation is about five orders of
magnitude lower than the string scale, $W_{I}$ should be generated at higher
(non-renormalisable) order, upon suitable vacuum expectation values (VEVs) of
fields that drive the theory ‘slightly’ away from the free-fermionic point.
Such VEVs are generated from an anomalous $U(1)_{A}$ which becomes massive by
absorbing the imaginary part of the dilaton (dual to the universal axion) that
transforms under the abelian gauge transformation cancelling the anomaly [21].
As a result, the $U(1)_{A}$ D-term is modified by a dilaton dependent constant
proportional to the one loop anomaly, imposing non-trivial VEVs for some
charged fields in order to satisfy the D-term vanishing condition. These VEVs
should also satisfy the F-flatness supersymmetry conditions and they are
naturally lower than the string scale by roughly a loop factor.

In this work, we perform the above described analysis by computing all
relevant non-renormalisable superpotential terms up to dimension eight,
suppressed by powers of the string scale $M_{s}^{n}$ with $n\leq 5$. We find a
consistent set of VEVs, satisfying all D and F-flatness conditions up to this
order, that give rise to the desired inflationary superpotential. It is
important to emphasise that this set of VEVs describe the inflationary phase
of the model, where in particular $SU(5)\times U(1)$ remains unbroken, and not
the vacuum after the end of inflation. Indeed, the $SU(5)\times U(1)$ breaking
occurs via a first order phase transition at a critical temperature which is
lower than the inflation scale [22]. As a result, the previous investigations
of the particle phenomenology of the model remain in principle valid, since
they describe the low energy vacuum.

The outline of the paper is the following. In Section 2, we recall for
convenience of the reader, the string construction of the ‘revamped’ flipped
$SU(5)\times U(1)$ model [1], its spectrum, the trilinear superpotential and
its main properties. In Section 3, we present a review of the Starobinsky-type
inflation within the context of no scale supergravity, in the particular form
emerging at the low energy limit of 4d free-fermionic heterotic superstrings.
In Section 4, we perform the computation and the analysis of the non-
renormalisable F-terms up to eighth order, we identify the inflaton and the
goldstino superfield and derive the desired inflationary superpotential. We
conclude in Section 5 with a summary of our results and outlook. Finally,
Appendix A contains the technical details of the string construction, while in
Appendix B we give some useful details of the calculation of the effective
field theory for inflation that we discuss in Section 3.

##  2 The “revamped” flipped $SU(5)\times U(1)$ string model

In this section, we give a short review of the flipped $SU(5)$ model [1]
constructed within the free-fermionic formulation of four-dimensional (4d)
heterotic strings [2]. For convenience of the reader, we use the same notation
as in the original paper [1]. The model is generated by a set of basis vectors
of boundary conditions for all world-sheet fermions with an appropriate choice
of mutual GSO projections, given for self-consistency in Appendix A. It is
equivalent to a $Z_{2}\times Z_{2}$ asymmetric orbifold of a $N=4$
supersymmetric model with gauge group $U(8)\times SO(2)^{6}\times SO(10)\times
SO(6)$, generated by the vectors
$\\{S,1,\zeta,\alpha,b_{1}+b_{4},b_{2}+b_{5}\\}$. The $Z_{2}\times Z_{2}$
operation defined by the vectors $\\{b_{1},b_{2}\\}$ breaks supersymmetry to
$N=1$, $U(8)\to U(5)\times U(1)^{3}$ and $SO(2)^{6}\to SO(2)$.

Thus, besides $SU(5)\times U(1)$, the gauge group contains a ‘hidden sector’
$SO(10)\times SO(6)$ and four abelian factors $U(1)^{4}$, one combination of
which has an anomaly cancelled by the Dine-Seiberg-Witten mechanism [21] and
acquires a mass upon absorbing the universal axion whose shift symmetry is
gauged under the anomalous $U(1)_{A}$ transformation. The matter spectrum
consists of:

  * •

Four chiral and one anti-chiral generations with the quantum numbers of quarks
and leptons, denoted as $(F_{i},\overline{f}_{i},\ell^{c}_{i})$ transforming
under $SU(5)$ as $({\bf 10},{\bf\bar{5}},{\bf 1})$ correspondingly and forming
$SO(10)$ spinors; two are coming from the first twisted plane $(i=1,4)$ in the
sectors $b_{1}$ and $b_{4}$, one from the third twisted plane $(i=3)$ in the
sector $b_{3}$, while the second twisted plane gives a pair of a chiral and
anti-chiral generations $(i=2,5)$ from the sectors $b_{2}$ and $b_{5}$,
respectively.

  * •

Four pairs of $({\bf 5}+{\bf\bar{5}})$ forming $SO(10)$ vectors and containing
electroweak higgses; three pairs are coming from the Neveu-Schwarz (NS)
$0$-sector $(h_{i},{\bar{h}}_{i})_{i=1,2,3}$ and one pair from the 3rd twisted
plane $(h_{45},{\bar{h}}_{45})$ from the sector $S+b_{4}+b_{5}$.

  * •

Five vector representations $({\bf 10},{\bf 1})+({\bf 1},{\bf 6})$ of the
hidden gauge group $SO(10)\times SO(6)$, denoted as $T_{i}$ and $D_{i}$,
respectively, in the sectors $b_{i}+2\alpha+\zeta$ and $b_{i}+2\alpha$; thus,
two are coming from the first twisted plane $(i=1,4)$, two from the second
$(i=2,5)$, and one from the third $i=3$.

  * •

Six pairs of $SO(6)$ spinors $({\bf 4}+{\bf\bar{4}})$ with fractional electric
charges $\pm 1/2$ that confine into integrally charged bound states; in their
notation below, $X,Y,Z$ correspond to the 1st, 2nd and 3rd twisted plane,
correspondingly.

  * •

Ten pairs of singlets under the observable and hidden gauge groups, but
charged under the four $U(1)$’s; seven are coming from the third twisted
sector $S+b_{4}+b_{5}$: $\phi_{1},\dots,\phi_{4}$, $\phi_{\pm}$, $\phi_{45}$
(and their conjugates $\bar{\phi}_{i}$, etc), while three are coming from the
NS sector: $\Phi_{23},\Phi_{31},\Phi_{12}$ (and their conjugates).

  * •

Five gauge singlets from the NS sector: $\Phi_{1},\dots,\Phi_{5}$.

The complete spectrum and quantum numbers are presented in Appendix A. The
full tree-level superpotential reads

$\displaystyle W_{3}=\,$ $\displaystyle g_{s}\sqrt{2}\left[\vrule
height=12.0pt,depth=1.0pt,width=0.0ptF_{1}F_{1}h_{1}+F_{2}F_{2}h_{2}+F_{4}F_{4}h_{1}+\overline{F}_{5}\overline{F}_{5}\overline{h}_{2}+F_{4}\overline{f}_{5}\overline{h}_{45}+F_{3}\overline{f}_{3}\overline{h}_{3}\right.$
$\displaystyle+\overline{f}_{1}\ell^{c}_{1}h_{1}+\overline{f}_{2}\ell^{c}_{2}h_{2}+\overline{f}_{5}\ell^{c}_{5}h_{2}+f_{4}\overline{\ell}^{c}_{4}\overline{h}_{1}$
$\displaystyle+\frac{1}{\sqrt{2}}F_{4}\overline{F}_{5}\phi_{3}+\frac{1}{\sqrt{2}}f_{4}\overline{f}_{5}\overline{\phi}_{2}+\frac{1}{\sqrt{2}}\overline{\ell}^{c}_{4}\ell^{c}_{5}\overline{\phi}_{2}$
$\displaystyle+h_{1}\overline{h}_{2}\overline{\Phi}_{12}+\overline{h}_{1}h_{2}\Phi_{12}+h_{2}\overline{h}_{3}\Phi_{23}+\overline{h}_{2}h_{3}\overline{\Phi}_{23}+h_{3}\overline{h}_{1}\overline{\Phi}_{31}$
$\displaystyle+\overline{h}_{3}h_{1}\Phi_{31}+h_{3}\overline{h}_{45}\overline{\phi}_{45}+\overline{h}_{3}h_{45}\phi_{45}+\frac{1}{2}h_{45}\overline{h}_{45}\Phi_{3}$
$\displaystyle+\phi_{1}\overline{\phi}_{2}\Phi_{4}+\overline{\phi}_{1}\phi_{2}\Phi_{4}+\phi_{3}\overline{\phi}_{4}\Phi_{5}+\overline{\phi}_{3}\phi_{4}\Phi_{5}+\frac{1}{2}\phi_{45}\overline{\phi}_{45}\Phi_{3}$
$\displaystyle+\frac{1}{2}\phi_{+}\overline{\phi}_{+}\Phi_{3}+\frac{1}{2}\phi_{-}\overline{\phi}_{-}\Phi_{3}$
(2.1)
$\displaystyle+\Phi_{12}\Phi_{23}\Phi_{31}+\overline{\Phi}_{12}\overline{\Phi}_{23}\overline{\Phi}_{31}+\Phi_{12}\phi_{+}\phi_{-}+\overline{\Phi}_{12}\overline{\phi}_{+}\overline{\phi}_{-}$
$\displaystyle+\frac{1}{2}\Phi_{3}\sum_{i=1}^{4}\phi_{i}\overline{\phi}_{i}+\Phi_{12}\sum_{i=1}^{4}\phi_{i}^{2}+\overline{\Phi}_{12}\sum_{i=1}^{4}\overline{\phi}_{i}^{2}$
$\displaystyle+D_{1}^{2}\overline{\Phi}_{23}+D_{2}^{2}\overline{\Phi}_{31}+D_{4}^{2}\overline{\Phi}_{23}+D_{5}^{2}\overline{\Phi}_{31}+\frac{1}{\sqrt{2}}D_{4}D_{5}\overline{\phi}_{3}$
$\displaystyle+T_{1}^{2}\overline{\Phi}_{23}+T_{2}^{2}\overline{\Phi}_{31}+T_{4}^{2}\overline{\Phi}_{23}+T_{5}^{2}\overline{\Phi}_{31}+\frac{1}{\sqrt{2}}T_{4}T_{5}\phi_{2}$
$\displaystyle+\frac{1}{\sqrt{2}}Y_{1}\overline{X}_{2}\phi_{4}+\frac{1}{\sqrt{2}}Y_{2}\overline{X}_{1}\phi_{1}+Y_{2}\overline{X}_{2}\phi_{+}+\frac{1}{2}Z_{1}\overline{Z}_{1}\Phi_{3}$
$\displaystyle\left.+Z_{2}\overline{Z}_{2}\overline{\Phi}_{12}+Z_{1}\overline{X}_{2}^{\prime}\ell^{c}_{2}+Y_{2}^{\prime}Z_{1}D_{1}\vrule
height=12.0pt,depth=1.0pt,width=0.0pt\right]\,,$

where $g_{s}$ is the string coupling. The breaking of the Grand Unified Theory
(GUT) group to the Standard Model (SM) is achieved by the VEVs
$\langle{\bar{F}}_{5}\rangle=\langle F\rangle$, where
$F\equiv\sum_{i=1}^{3}\alpha_{i}F_{i}$, along a supersymmetric F-flat
direction that leaves the coefficients $\alpha_{i}$ of the linear combination
$F$ undetermined [1]. These are accompanied by a set of VEVs for $SU(5)\times
U(1)$ singlets subject to F- and D-term vanishing conditions, where in the
former higher order (non-renormalisable) terms play in principle an important
rôle. The phenomenology of the model, including quark masses and mixings,
baryon decay and neutrino masses has been studied extensively in the past [1,
3, 4, 5, 6]. The effective field theory of the model is a $N=1$ no-scale
supergravity [14] with a Kähler potential of chiral multiplets that can be
computed exactly at the string tree-level (to all orders in $\alpha^{\prime}$)
using the method of Refs. [15, 16, 17], applied for generic 4d free-fermionic
heterotic superstrings.

##  3 Starobinsky type inflation from no-scale supergravity

It is shown that Starobinsky type inflation [7] can be easily embedded in no-
scale supergravity implemented with an appropriate superpotential for the
inflaton field [8, 9, 10, 11, 12, 13]. Here we will use a particular proposal
[9, 18, 19, 20] that can be naturally derived in the effective field theory of
free-fermionic heterotic constructions.

The Starobinsky model of inflation is obtained by adding to the Einstein
action a quadratic in the scalar curvature term. This can be linearised by
introducing a Lagrange multiplier which becomes, in the Einstein frame, an
ordinary scalar field (the so-called scalaron) with a particular potential
[23] that can drive inflation compatible with present cosmological
observations. Since the scalar curvature belongs in supergravity to a chiral
multiplet [24], the supersymmetric extension of $R^{2}$ must be written as a
D-term, which needs two superfield-Lagrange multipliers to be linearised [25].
One of them contains the inflaton, while the other is the goldstino superfield
that breaks supersymmetry. The model is described by a no-scale type
supergravity with a Kähler potential $K$ and a superpotential $W$ given by:

$K=-3\ln\left(T+{\bar{T}}-C{\bar{C}}\right)\quad;\quad W=\mu C(T-{k})\,,$
(3.1)

where $\mu$ is the inflation scale and $k$ is an irrelevant constant that can
be set to 1 by an appropriate rescaling of the fields ($T$ by $k$ and $C$ by
$k^{1/2}$). The normalised inflaton $\varphi$ is given by ${\rm
Re}T=e^{\sqrt{2/3}\varphi}$, while the linear term in $C$ of the
superpotential drives the supersymmetry breaking.

Since in the free-fermionic constructions all moduli (except the dilaton) are
fixed to particular values in general of enhanced symmetries (gauge or
discrete), it is convenient to go to an appropriate basis where fields have
vanishing VEVs, describing fluctuations around the free-fermionic point.
Defining

$T={1+y\over 1-y}\quad;\quad C=\sqrt{2}{z\over 1-y}\,,$ (3.2)

one obtains:

$K=-3\ln\left(1-|y|^{2}-|z|^{2}\right)\quad;\quad W=\mu
z\left(y-y^{2}\right)\,,$ (3.3)

upon a suitable Kähler transformation. The inflation region corresponds to
$\varphi\to\infty$, or equivalently $|y|\to 1$, for vanishing $C$ or $z$,
where the scalar potential is approximately constant $\propto\mu^{2}$. The
main problem of this model is that the sgoldstino $C$ (or $z$) is unstable
(tachyonic) during inflation [10]. Stabilisation of the runaway direction can
be achieved by adding for instance a quartic (or higher order) term in the
Kähler potential which obviously modifies the theory beyond the minimal
supersymmetrisation of $R^{2}$. Alternatively, one can parametrise the
stabilisation mechanism in a model independent way by integrating out the
heavy sgoldstino replacing $C$ with the nilpotent goldstino superfield
satisfying the constraint $C^{2}=0$ [12]. Here, we will follow a different
approach based on the derived effective field theory of free-fermionic string
models.

The first observation is that the Kähler potential is in general a sum of
different factors, splitting the logarithm (3.3) into pieces. When twisted
fields are set to zero, the Kähler potential of all untwisted fields splits in
four pieces:

$\displaystyle K_{UT}=K_{0}+\sum_{i=1}^{3}K_{i}\quad;\quad K_{0}$
$\displaystyle=$ $\displaystyle-\ln(S+{\bar{S}})$ (3.4) $\displaystyle K_{i}$
$\displaystyle=$
$\displaystyle-\ln\left(1-\sum_{A=1}^{n_{i}}\left|\Phi_{i,A}\right|^{2}+{1\over
4}\left|\sum_{A=1}^{n_{i}}\Phi_{i,A}^{2}\right|^{2}\right)$

where $S$ denotes the dilaton superfield and $K_{i}$ parametrise an
$SO(2,n_{i})/SO(2)\times SO(n_{i})$ manifold [15, 16]. As we mentioned in the
previous section, a $N=1$ supersymmetric free-fermionic construction
corresponds to a $Z_{2}\times Z_{2}$ asymmetric orbifold acting on a $N=4$
theory. The orbifold group has therefore three $Z_{2}$ non-trivial elements
$h_{i}$, $i=1,2,3$. The action of every element alone gives rise to a $N=2$
theory. The set of the $n_{i}$ chiral superfields $\\{\Phi_{i,A}\\}$
corresponds to those coming from the vector multiplets of the $i$-th $N=2$
theory surviving the other two projections. In the weak field limit, twisted
fields give an additional contribution to the Kähler potential (3.4):

$K_{T}=\sum_{i=1}^{3}K_{T_{i}}\quad;\quad
K_{T_{i}}\simeq\sum_{\\{\Phi_{T_{i}}\\}}|\Phi_{T_{i}}|^{2}e^{K_{j}+K_{l}}\,,\,i\neq
j\neq l\neq i$ (3.5)

where the sum is extended over all chiral superfields of the $i$-th twisted
sector. On the other hand, in the case one keeps the fields of a single
twisted sector, say $T_{1}$, $K_{T_{1}}$ becomes to all orders [16]:

$K_{T_{1}}=-2\ln\left(1-{1\over
2}\sum_{\\{\Phi_{T_{1}}\\}}|\Phi_{T_{1}}|^{2}e^{{1\over
2}(K_{2}+K_{3})}\right)\,.$ (3.6)

Adding $K_{2}+K_{3}$ from (3.4) to $K_{T_{1}}$, one thus have

$K_{2}+K_{3}+K_{T_{1}}=-2\ln\left(e^{-{1\over 2}(K_{2}+K_{3})}-{1\over
2}\sum_{\\{\Phi_{T_{1}}\\}}|\Phi_{T_{1}}|^{2}\right)\,.$ (3.7)

In the proposal [22], the inflaton $y$ was identified with the superpartner of
a state that mixes with a R-handed neutrino entering in the so-called
$\lambda_{6}$ coupling of the flipped $SU(5)$ superpotential [1] that provides
neutrino masses upon the GUT symmetry breaking. The inflaton can then decay in
R-handed neutrinos reheating the universe and there is an interesting
connection with the neutrino mass generation. This possibility is compatible
with the GUT symmetry breaking occurring by a first order transition at a
critical temperature which is lower than the inflation scale [22]. As a
result, $SU(5)\times U(1)$ remains unbroken during the inflationary era.

In the model described in the previous section, the $\lambda_{6}$ coupling
corresponds to the term $F_{4}{\bar{F}}_{5}\phi_{3}$ in the superpotential
(2.1). The inflaton $y$ should then correspond to a linear combination of
fields containing $\phi_{3}$ that originates from the third twisted sector
$T_{3}$. By inspection of the Kähler potentials (3.4) - (3.7), we then
consider the possibility that the goldstino superfield $z$ appearing in the
inflationary superpotential (3.1) comes from the third untwisted sector, so
that its corresponding Kähler potential $K_{3}$ does not appear inside the
logarithm (3.6). Using $K_{3}$ from (3.4) for the case of one field $z$
parametrising an $SO(2,1)/SO(2)$ manifold and restricting (3.7) to a single
field $y\equiv\Phi_{T_{3}}/\sqrt{2}$, one obtains:

$\displaystyle K$
$\displaystyle=-\ln(S+{\bar{S}})-2\ln\left(1-\frac{1}{2}|z|^{2}\right)-2\ln\left(1-\left|y\right|^{2}\right)$
(3.8) $\displaystyle W$ $\displaystyle=\mu z\left(y-y^{2}\right)$

where all other fields are set to zero, corresponding to the fermionic point;
VEVs away from it are treated perturbatively.

Defining $y=\rho e^{i\theta}$ with $0\leq\rho<1$, the resulting scalar
potential reads:

$\displaystyle V$
$\displaystyle\\!=\\!e^{K}\left(\sum_{\Phi=S,y,z}\left|D_{\Phi}W\right|^{2}-3\left|W\right|^{2}\right)$
(3.9)
$\displaystyle\\!=\\!\mu^{2}g_{s}^{2}\\!\left\\{\\!\frac{4\rho^{2}+2(1\\!-\\!2\rho\\!-\\!\rho^{2})\left|z\right|^{2}+\rho^{2}\left|z\right|^{4}}{(1+\rho)^{2}\left(2-\left|z\right|^{2}\right)^{2}}\\!+\\!4\frac{4\rho^{2}+4|z|^{2}+\rho^{2}|z|^{4}}{\left(1-\rho^{2}\right)^{2}\left(2-\left|z\right|^{2}\right)^{2}}\rho\sin^{2}{\theta\over
2}\\!\right\\},$

where $D_{\Phi}\equiv\partial_{\Phi}+(\partial_{\Phi}K)$ is the Kähler
covariant derivative and $g_{s}$ is the string coupling which we consider
constant assuming a suitable stabilisation mechanism for the dilaton.
Minimisation with respect to the phase gives $\theta=0$ and the potential
becomes:

$V{\Large\mid}_{\theta=0}=\frac{\mu^{2}g_{s}^{2}}{(1+\rho)^{2}}\left(\rho^{2}+\frac{1}{2}\frac{(1-\rho)^{2}\left|z\right|^{2}}{\left(1-\frac{\left|z\right|^{2}}{2}\right)^{2}}\right)\;.$
(3.10)

It is easy to see that the phase fluctuation acquires a mass of order the
inflation scale and decouples during inflation. We also find very interesting
that the sgoldstino direction $z$ is not any more unstable during inflation.
The dilaton contribution raised the tachyonic mass to a positive value so that
$z$ becomes asymptotically a flat direction in the limit $\rho\to 1$. More
precisely, in the absence of the dilaton, the factor $(1-\rho)^{2}$
multiplying $|z|^{2}$ in the expression (3.10) becomes
$(1-\rho)^{2}-2\rho^{2}$.

In order to study the supersymmetry breaking, one should compute the values of
the auxiliary fields $F_{\Phi}=e^{K/2}D_{\Phi}W$ for all fields $\Phi$. It is
easy to see that all of them vanish at $\theta=z=0$ except of $F_{z}$:

$F_{y}{\Large\mid}_{z=\theta=0}=F_{S}{\Large\mid}_{z=\theta=0}=0\quad;\quad
F_{z}{\Large\mid}_{z=\theta=0}=\mu g_{s}\frac{\rho}{1+\rho}\,.$ (3.11)

It follows that, as expected, $z$ plays the rôle of the goldstino superfield
that breaks spontaneously supersymmetry during inflation at a scale $\mu
g_{s}/2$.

The canonically normalised inflaton field $\varphi$ is given by:

$\varphi=\ln\frac{1+\rho}{1-\rho}\quad;\quad\rho=\tanh{\varphi\over 2}\,,$
(3.12)

and the potential at $z=0$ becomes

$V{\Large\mid}_{z=\theta=0}=\mu^{2}g_{s}^{2}\left(\frac{\rho}{1+\rho}\right)^{2}=\frac{\mu^{2}g_{s}^{2}}{4}\left(1-e^{-\varphi}\right)^{2}\,.$
(3.13)

For details see Appendix B. It has the same form as in the case $\alpha=2/3$
of Ref. [19] from which one can deduce the inflationary predictions of the
model for the spectral index $n_{s}$ of primordial density perturbations and
the ratio $r$ of tensor-to-scalar modes:

$n_{s}=0.965\quad;\quad r=2.4\times 10^{-3}\,,$ (3.14)

for the canonical value of about $N_{\ast}=55$ e-folds of inflation.

In the next section, we derive the model we described above in string theory
within the “revamped” flipped $SU(5)\times{U(1)}_{A}$ construction we
described in Section 2. As we discussed already, the Kähler potential (3.8) is
fixed by an appropriate choice of fields and the main goal of the analysis
will be to compute the inflationary superpotential. Our strategy is to
generate it at a higher non-renormalisable level, via VEVs of $SU(5)\times
U(1)$ singlet fields, since this gauge symmetry remains unbroken at the
inflation scale. Non-trivial VEVs are driven by the anomalous $U(1)_{A}$
cancellation mechanism and are naturally a loop factor below the string scale.
One may also worry on the relative coefficient ($-1$ in our basis) between the
quadratic and cubic term in $W$, in (supergravity) Planck units, implying in
general a fine-tuning of the corresponding VEVs. Replacing $-1$ by an order
one parameter $-\lambda$:

$W=\mu z\left(y-\lambda y^{2}\right)$ (3.15)

(the sign is not relevant since it can change by redefining $y\to-y$), the
potential will diverge at the boundary $y=1$ but an inflationary period of
about 60 e-folds remains present for a range of values of $\lambda$ around
unity within about $10^{-3}$ [20].

##  4 Realisation of a Starobinsky-like inflationary scenario in the flipped
${SU(5)}\times{U(1)}$ string model

In this section we consider a concrete realisation of the Starobinsky-like
inflationary scenario described in Section 3 in the context of the “revamped”
${SU(5)}\times{U(1)}$ string model presented in Section 2. In this framework,
a central rôle is played by the so called $\lambda_{6}$ superpotential
coupling, that is the coupling relating a gauge singlet, the GUT breaking
Higgs and matter fields, which eventually provides the neutrino see-saw
mechanism. In the flipped $SU(5)$ string model under consideration there
exists a single term of this type in the tree-level superpotential (2.1),
namely $F_{4}\overline{F}_{5}\phi_{3}$, where $F_{4}$ is a matter field and
$\overline{F}_{5}$ is the flipped $SU(5)$ breaking Higgs field. Following
Section 3 the singlet $\phi_{3}$ has to be linked to the inflaton field. This
in turn requires $\phi_{3}$ to remain massless at leading order in the
superpotential, whereas it may acquire Majorana mass and mixings with other
singlet field(s) at some higher order. As a matter of fact, intermediate mass
scales can be generated in the context of low energy string effective theories
via the use of non-renormalisable interactions, as follows. In the presence of
an anomalous abelian symmetry D-flatness requires some fields in the theory to
develop VEVs one or two orders of magnitude below the Planck scale. These in
turn can entail additional field VEVs via the F-flatness conditions.

A superpotential term of the form $\phi_{1}\phi_{2}\phi_{3}\varphi^{N-3}$,
where $\varphi^{N-3}$ stands for a product of $N-3$ fields acquiring VEVs,
gives rise to an effective interaction $\lambda\phi_{1}\phi_{2}\phi_{3}$, with
$\lambda\sim\left(\frac{\langle\varphi\rangle}{M_{P}}\right)^{N-3}$. However,
these non-renormalisable terms are subject to non-trivial string selection
rules that go beyond mere gauge group invariance [26, 27]. For example, a pure
Neveu–Schwarz (NS) coupling, that is a coupling consisting exclusively of
fields arising from the $0/S$ sector (vectors of boundary conditions),
vanishes identically for any $N>3$. More particularly, it can be shown, that
in the model under consideration, all non-renormalisable interactions
involving solely singlet fields vanish identically [26]. This stems from the
fact that in the specific model non-abelian gauge group singlets come solely
from two sectors, the NS $0/S$ sector and the third twisted sector $T_{3}$ of
the $b_{4}+b_{5}/S+b_{4}+b_{5}$ vectors of boundary conditions. It follows
that generation of a non-trivial bilinear or trilinear superpotential
involving $\phi_{3}$ at a higher order, requires the introduction of
additional VEVs for some hidden sector fields (twisted in the first and/or the
second plane). In view of this requirement, and in order to examine the
feasibility of Starobinsky-like inflationary scenario, we start by deriving
the constraints on all possible VEV assignments for $SU(5)\times U(1)$ gauge
singlet fields that may transform non-trivially under the hidden group
$SO(10)\times SO(6)$.

The revamped flipped string model gauge symmetry is
$[SU(5)\times{U(1)}]\times{U(1)}^{4}\times{[SO(10)}\times{SU(4)]}$. As
explained in Appendix A, the abelian factor product ${U(1)}^{4}$ can be recast
in the form
$\left\\{\prod_{I=1}^{3}{U(1)}_{I}^{\prime}\right\\}\times{U(1)}^{\prime}_{A}$,
where $U(1)^{\prime}_{I},I=1,2,3$ are anomaly free and $U(1)^{\prime}_{A}$ is
anomalous. The anomalous abelian symmetry is broken via the Dine-Seiberg-
Witten mechanism at the loop level [21], involving therefore a VEV of the
dilaton field, i.e. the string coupling. The induced Fayet-Iliopoulos term
breaks supersymmetry unless it is cancelled by a VEV along an appropriate
matter field combination. In this case, the D-flatness conditions associated
with the ${U(1)^{\prime}}^{4}$ group factor read [1, 28]:

$\displaystyle{\cal D}_{I}$
$\displaystyle=\sum_{i}q^{i}_{I}\left|\varphi_{i}\right|^{2}=0\,,I=1,2,3\,,$
(4.1) $\displaystyle{\cal D}_{A}$
$\displaystyle=\sum_{i}q^{i}_{A}\left|\varphi_{i}\right|^{2}+\frac{1}{192\pi^{2}}{2\over\alpha^{\prime}}\,{\rm
Tr}Q_{A}=0\,,$ (4.2)

where $\varphi_{i}$ stand for massless fields with charges
$q^{i}_{I}/q^{i}_{A}$ under ${U(1)^{\prime}}_{I}/{U(1)^{\prime}_{A}}$ and the
trace of the anomalous $U(1)^{\prime}_{A}$ generator ${\rm Tr}{Q_{A}}=180$. In
addition, $N=1$ supersymmetry imposes the F-flatness constraints

$\displaystyle{\cal F}_{i}=\frac{\partial W}{\partial\varphi_{i}}=0\,,$ (4.3)

where $W$ stands for the superpotential.

For our purposes, we will assume VEVs for the singlet fields
$\Phi_{A},\overline{\Phi}_{A},A=12,23,13\,$,
$\Phi_{i},i=1,\dots,5,\,\phi_{j},\overline{\phi}_{j},j=1,2,3,4,+,-,45$ as well
as the “hidden” sector fields $D_{a},T_{a}\,a=1,\dots,5$. Following [4], the
D-flatness conditions (4.1) and (4.2) can be recast in the form:

$\displaystyle\left|\phi_{45}\right|^{2}-\left|\overline{\phi}_{45}\right|^{2}-\frac{1}{2}\left(\left|D_{3}\right|^{2}+\left|T_{3}\right|^{2}\right)=\xi^{2}\,,$
(4.4)
$\displaystyle\left|\phi_{+}\right|^{2}-\left|\phi_{-}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}+\left|\overline{\phi}_{-}\right|^{2}+\frac{1}{2}\left(\left|D_{3}\right|^{2}-\left|T_{3}\right|^{2}\right)=\xi^{2}\,,$
(4.5)
$\displaystyle\left|\Phi_{31}\right|^{2}-\left|\overline{\Phi}_{31}\right|^{2}-\left|\Phi_{23}\right|^{2}+\left|\overline{\Phi}_{23}\right|^{2}-\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{2}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}-\left|D_{5}\right|^{2}\right)$
$\displaystyle\
-\frac{1}{2}\left(\left|T_{1}\right|^{2}+\left|T_{2}\right|^{2}+\left|T_{3}\right|^{2}-\left|T_{4}\right|^{2}+\left|T_{5}\right|^{2}\right)=3\xi^{2}\,,$
(4.6)
$\displaystyle\left|\Phi_{23}\right|^{2}-\left|\overline{\Phi}_{23}\right|^{2}-\left|\Phi_{12}\right|^{2}+\left|\overline{\Phi}_{12}\right|^{2}+\frac{1}{2}\sum_{i=1}^{4}\left(\left|\phi_{i}\right|^{2}-\left|\overline{\phi}_{i}\right|^{2}\right)+\left|\phi_{+}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}$
$\displaystyle\
+\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}\right)+\frac{1}{2}\left(\left|T_{1}\right|^{2}-\left|T_{4}\right|^{2}\right)=0\,,$
(4.7)

where

$\displaystyle\xi^{2}=\frac{1}{16\pi^{2}}{2\over\alpha^{\prime}}\,.$ (4.8)

The F-flatness conditions can be computed utilizing (4.3). Restricting to the
tree-level superpotential contributions (2.1) we have

$\displaystyle\left({\Phi_{12}}\right)$
$\displaystyle:\sum_{i=1}^{4}\phi_{i}^{2}+\Phi_{23}\Phi_{31}+\phi_{+}\phi_{-}=0\,,$
(4.9) $\displaystyle\left(\overline{\Phi}_{12}\right)$
$\displaystyle:\sum_{i=1}^{4}\overline{\phi}_{i}^{2}+\overline{\Phi}_{23}\overline{\Phi}_{31}+\overline{\phi}_{+}\overline{\phi}_{-}=0\,,$
(4.10) $\displaystyle\left(\Phi_{31}\right)$
$\displaystyle:\Phi_{12}\Phi_{23}+D_{2}^{2}+T_{2}^{2}+T_{5}^{2}=0\,,$ (4.11)
$\displaystyle\overline{\Phi}_{31}$
$\displaystyle:\overline{\Phi}_{12}\overline{\Phi}_{23}+D_{5}^{2}=0\,,$ (4.12)
$\displaystyle\left({\Phi}_{23}\right)$
$\displaystyle:{\Phi}_{12}{\Phi}_{31}+T_{4}^{2}=0\,,$ (4.13)
$\displaystyle\left(\overline{\Phi}_{23}\right)$
$\displaystyle:\overline{\Phi}_{12}\overline{\Phi}_{31}+D_{1}^{2}+D_{4}^{2}+T_{1}^{2}=0\,,$
(4.14) $\displaystyle\left(\Phi_{3}\right)$
$\displaystyle:\sum_{i=1}^{4}\phi_{i}\overline{\phi}_{i}+\phi_{45}\overline{\phi}_{45}+\phi_{-}\overline{\phi}_{-}+\phi_{+}\overline{\phi}_{+}=0\,,$
(4.15) $\displaystyle\left(\Phi_{4}\right)$
$\displaystyle:\phi_{2}\overline{\phi}_{1}+\phi_{1}\overline{\phi}_{2}=0\,,$
(4.16) $\displaystyle\left(\Phi_{5}\right)$
$\displaystyle:\phi_{4}\overline{\phi}_{3}+\phi_{3}\overline{\phi}_{4}=0\,,$
(4.17) $\displaystyle\left(\phi_{1}\right)$
$\displaystyle:2\phi_{1}\Phi_{12}+\Phi_{3}\overline{\phi}_{1}+\Phi_{4}\overline{\phi}_{2}=0\,,$
(4.18) $\displaystyle\left(\phi_{2}\right)$
$\displaystyle:2\phi_{2}\Phi_{12}+\Phi_{4}\overline{\phi}_{1}+\Phi_{3}\overline{\phi}_{2}+T_{4}{\cdot}T_{5}=0\,,$
(4.19) $\displaystyle\left(\phi_{3}\right)$
$\displaystyle:2\phi_{3}\Phi_{12}+\Phi_{3}\overline{\phi}_{3}+\Phi_{5}\overline{\phi}_{4}=0\,,$
(4.20) $\displaystyle\left(\phi_{4}\right)$
$\displaystyle:2\phi_{4}\Phi_{12}+\Phi_{5}\overline{\phi}_{3}+\Phi_{3}\overline{\phi}_{4}=0\,,$
(4.21) $\displaystyle\left(\overline{\phi}_{1}\right)$
$\displaystyle:2\overline{\phi}_{1}\overline{\Phi}_{12}+\Phi_{3}{\phi}_{1}+\Phi_{4}{\phi}_{2}=0\,,$
(4.22) $\displaystyle\left(\overline{\phi}_{2}\right)$
$\displaystyle:2\overline{\phi}_{2}\overline{\Phi}_{12}+\Phi_{4}{\phi}_{1}+\Phi_{3}{\phi}_{2}=0\,,$
(4.23) $\displaystyle\left(\overline{\phi}_{3}\right)$
$\displaystyle:2\overline{\phi}_{3}\overline{\Phi}_{12}+\Phi_{3}{\phi}_{3}+\Phi_{5}{\phi}_{4}+D_{4}{\cdot}D_{5}=0\,,$
(4.24) $\displaystyle\left(\overline{\phi}_{4}\right)$
$\displaystyle:2\overline{\phi}_{4}\overline{\Phi}_{12}+\Phi_{5}{\phi}_{3}+\Phi_{3}{\phi}_{4}=0\,,$
(4.25) $\displaystyle\left(\phi_{+}\right)$
$\displaystyle:\Phi_{3}\overline{\phi}_{+}+\Phi_{12}\phi_{-}=0\,,$ (4.26)
$\displaystyle\left(\overline{\phi}_{+}\right)$
$\displaystyle:\Phi_{3}\overline{\phi}_{-}+\overline{\Phi}_{12}\phi_{+}=0\,,$
(4.27) $\displaystyle\left(\phi_{-}\right)$
$\displaystyle:\Phi_{3}\overline{\phi}_{-}+\Phi_{12}\phi_{+}=0\,,$ (4.28)
$\displaystyle\left(\overline{\phi}_{-}\right)$
$\displaystyle:\Phi_{3}\overline{\phi}_{+}+\overline{\Phi}_{12}\phi_{-}=0\,,$
(4.29) $\displaystyle\left(\phi_{45}\right)$
$\displaystyle:\Phi_{3}\overline{\phi}_{45}=0\,,$ (4.30)
$\displaystyle\left(\overline{\phi}_{45}\right)$
$\displaystyle:\Phi_{3}\phi_{45}=0\,,$ (4.31)
$\displaystyle\left(T_{1}\right)$
$\displaystyle:2\overline{\Phi}_{23}T_{1}=0\,,$ (4.32)
$\displaystyle\left(T_{2}\right)$ $\displaystyle:2\Phi_{31}T_{2}=0\,,$ (4.33)
$\displaystyle\left(T_{4}\right)$
$\displaystyle:2\Phi_{23}T_{4}+\phi_{2}T_{5}=0\,,$ (4.34)
$\displaystyle\left(T_{5}\right)$
$\displaystyle:\phi_{2}T_{4}+2\Phi_{31}T_{5}=0\,,$ (4.35)
$\displaystyle\left(D_{1}\right)$
$\displaystyle:2D_{1}\overline{\Phi}_{23}=0\,,$ (4.36)
$\displaystyle\left(D_{2}\right)$ $\displaystyle:2D_{2}{\Phi}_{31}=0\,,$
(4.37) $\displaystyle\left(D_{4}\right)$
$\displaystyle:2D_{4}\overline{\Phi}_{23}+D_{5}\overline{\phi}_{3}=0\,,$
(4.38) $\displaystyle\left(D_{5}\right)$
$\displaystyle:2D_{5}\overline{\Phi}_{31}+D_{4}\overline{\phi}_{3}=0\,,\hskip
85.35826pt$ (4.39)

where $(\varphi_{i})$ denotes the flatness equation corresponding to
${\partial W}/{\partial\varphi_{i}}$.

A solution of the F/D-flatness equations (4.9)-(4.39) involves, in general, a
set of fields with vanishing VEVs and a set of fields with VEVs of order
$\xi$. For the purpose of our study and after trial and error analysis, we
will focus on the subspace of solutions defined by

$\displaystyle\Phi_{12}=\overline{\Phi}_{12}=\Phi_{23}=\overline{\Phi}_{23}=\overline{\Phi}_{31}=0\,,$
$\displaystyle\Phi_{3}=\Phi_{4}=\Phi_{5}=0\,,$
$\displaystyle\phi_{1}=\phi_{2}=\phi_{3}=\overline{\phi}_{1}=\overline{\phi}_{2}=\overline{\phi}_{3}=0\,,$
(4.40) $\displaystyle\overline{F}_{5}=F_{1}=F_{2}=F_{3}=F_{4}=0\,,$
$\displaystyle D_{2}=T_{2}=T_{5}=0\,,$ $\displaystyle
T_{4}^{2}=D_{5}^{2}=D_{4}{\cdot}D_{5}=0\,.$

Note that $T_{a}$/$D_{a}$ are complex $SO(10)$/$SO(6)$ vectors, and thus, they
can be nilpotent without being vanishing. The above choice has the advantage
of reducing the tree-level F-flatness equations (4.9)-(4.39) to

$\displaystyle(\Phi_{12})$ $\displaystyle:\phi_{4}^{2}+\phi_{+}\phi_{-}=0\,,$
(4.41) $\displaystyle(\overline{\Phi}_{12})$
$\displaystyle:\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}=0\,,$
(4.42) $\displaystyle\left(\overline{\Phi}_{23}\right)$
$\displaystyle:D_{1}^{2}+D_{4}^{2}+T_{1}^{2}=0\,,$ (4.43)
$\displaystyle\left(\Phi_{3}\right)$
$\displaystyle:\phi_{4}\overline{\phi}_{4}+\phi_{45}\overline{\phi}_{45}+\phi_{+}\overline{\phi}_{+}+\phi_{-}\overline{\phi}_{-}=0\,.$
(4.44)

In addition, the D-flatness conditions (4.4)-(4.5) remain unchanged while
(4.6)-(4.7) now read

$\displaystyle\left|\Phi_{31}\right|^{2}-\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}-\left|D_{5}\right|^{2}\right)-\frac{1}{2}\left(\left|T_{1}\right|^{2}+\left|T_{3}\right|^{2}-\left|T_{4}\right|^{2}\right)=3\xi^{2}\,,$
(4.45)
$\displaystyle\left|\phi_{4}\right|^{2}-\left|\overline{\phi}_{4}\right|^{2}+2\left(\left|\phi_{+}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}\right)+\left|D_{1}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}+\left|T_{1}\right|^{2}-\left|T_{4}\right|^{2}=0\,.$
(4.46)

Consequently, the solution (4.40) has enough free parameters as to allow
compliance with the D-flatness requirements.

Let us now turn to the tree-level singlet field masses. A detailed calculation
shows that any solution involving non-vanishing VEVs for all singlet fields
not included in (4.40) renders
$\Phi_{12},\overline{\Phi}_{12},\Phi_{3},\Phi_{5}$ as well as four linear
combinations of
$\phi_{3},\overline{\phi}_{3},\phi_{4},\overline{\phi}_{4},\phi_{+},\overline{\phi}_{+},\phi_{-},\overline{\phi}_{-},\phi_{45},\overline{\phi}_{45}$,
$\Phi_{23}$ superheavy. Focusing on mass-mixings involving the field
$\phi_{3}$, we have only two relevant superpotential terms

$\displaystyle
w_{3m}\sim\Phi_{5}\left(\phi_{3}\overline{\phi}_{4}+\phi_{4}\overline{\phi}_{3}\right)$
(4.47)

that result to a massless $\phi_{0}$ and a massive $\phi_{m}$ linear
combinations

$\displaystyle\phi_{0}=\sin\omega\,\phi_{3}-\cos\omega\,\overline{\phi}_{3}\,,$
(4.48)
$\displaystyle\phi_{m}=\cos\omega\,\phi_{3}+\sin\omega\,\overline{\phi}_{3}\,,$
(4.49)

where
$\tan\omega={\langle\phi_{4}\rangle}/{\langle\overline{\phi}_{4}\rangle}$. In
view of this, we identify the inflaton field $y$ with the massless combination
$\phi_{0}$. Solving backwards, we find that the inflaton participates in
interactions via the fields

$\displaystyle\phi_{3}=\sin\omega\,\phi_{0}+\dots\quad;\quad\overline{\phi}_{3}=-\cos\omega\,\phi_{0}+\dots$
(4.50)

where the dots stand for the superheavy combination $\phi_{m}$.

Of key importance to our analysis are the higher order non-renormalisable (NR)
corrections to the superpotential (2.1). A generic non-renormalisable term at
order $N>3$ is of the form

$\displaystyle
g_{s}^{N-2}C_{N}\,\varphi_{1}\varphi_{2}\varphi_{3}\dots\varphi_{N}\,,$ (4.51)

in units $\sqrt{2\alpha^{\prime}}=1$, where $C_{N}$ is a numerical constant
proportional to the correlator

$\displaystyle\langle\varphi_{1}^{F}\varphi_{2}^{F}\varphi_{3}^{B}\dots\varphi_{N}^{B}\rangle\,.$
(4.52)

Here, $\varphi_{i}^{F}/\varphi_{i}^{B}$ stand for the fermionic/bosonic vertex
operators of the superfield $\varphi_{i}$. The constant $C_{N}$ can be
computed explicitly and it turns out to be either zero or of order one [29,
30],[28].111Actually $C_{3}=\sqrt{2}$, while $C_{N}$ with $N\geq 4$ was found
to be of order 10 for the particular examples it was computed. The vanishing
of $C_{N}$ is controlled by gauge invariance, Ising correlators related to
real fermionic internal coordinates, and additional selection rules ascribed
to $N=2$ worldsheet superalgebra[27]. As the number of candidate NR terms is
growing fast with the order $N$, a systematic application of all selection
criteria is only possible via the use of a computer code. In our analysis we
have used a specialised computer program to calculate all non-vanishing NR
contributions to the superpotential up to sixth order.

To order $N=6$ the F-flatness conditions, after applying (4.40) and choosing

$\displaystyle\Phi_{2}=T_{1}^{2}=T_{3}^{2}=T_{3}{\cdot}T_{4}=D_{3}=D_{1}^{2}=D_{4}^{2}=0\,,$
(4.53)

read

$\displaystyle\left(\Phi_{12}\right)$
$\displaystyle:\phi_{4}^{2}+\phi_{+}\phi_{-}+\left\\{\left(D_{1}{\cdot}D_{5}\right)^{2}+\left(D_{4}{\cdot}D_{5}\right)^{2}\right\\}=0\,,$
(4.54) $\displaystyle(\overline{\Phi}_{12})$
$\displaystyle:\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}=0\,,$
(4.55) $\displaystyle\left(\Phi_{3}\right)$
$\displaystyle:\phi_{4}\overline{\phi}_{4}+\phi_{45}\overline{\phi}_{45}+\phi_{+}\overline{\phi}_{+}+\phi_{-}\overline{\phi}_{-}=0\,,$
(4.56) $\displaystyle\left(\overline{\phi}_{3}\right)$
$\displaystyle:D_{4}{\cdot}D_{5}+\left\\{\left(D_{1}{\cdot}D_{5}\right)\left(T_{1}{\cdot}T_{4}\right)\right\\}=0\,,$
(4.57) $\displaystyle\left(T_{1}\right)$
$\displaystyle:\left\\{\left(\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\Phi_{31}T_{1}\right\\}=0\,,$
(4.58) $\displaystyle\left(D_{1}\right)$
$\displaystyle:\left\\{\left(\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\Phi_{31}D_{1}\right\\}=0\,,$
(4.59) $\displaystyle\left(D_{4}\right)$
$\displaystyle:\left\\{\left(\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\Phi_{31}D_{4}\right\\}=0\,,$
(4.60)

where we have omitted for notational simplicity order one numerical
coefficients for terms arising from NR interactions (terms in curly brackets).
It is possible to extend the tree-level solution (4.40), (4.41),(4.44) as to
include up to order $N=6$ NR corrections. To this end, we remark that $\xi$
defined in (4.8) is the only explicit parameter in the system of flatness
constraints. In addition, $\xi$ is of the order of $10^{-1}-10^{-2}$ in string
units. Hence, we can consider solutions expressed as power series in the small
parameter $\xi$, starting from a tree-level expression solving Eq. (4.44) at
the leading order in this series expansion. In this regard, we can solve the
$\left(\overline{\phi}_{3}\right)$ flatness (4.57) assuming
$D_{4}{\cdot}D_{5}$ is of the order $\xi^{4}$

$\displaystyle
D_{4}{\cdot}D_{5}=-\left\\{\left(D_{1}{\cdot}D_{5}\right)\left(T_{1}{\cdot}T_{4}\right)\right\\}\sim\xi^{4}\,.$
(4.61)

Then, we can also solve Eq. (4.54) perturbatively

$\displaystyle\phi_{4}^{2}+\phi_{+}\phi_{-}=\left\\{\left(D_{1}{\cdot}D_{5}\right)^{2}\right\\}\sim\xi^{4}\,.$
(4.62)

To solve Eqs. (4.58)-(4.60) together with (4.55) we also choose

$\displaystyle\overline{\phi}_{4}^{2}=-\overline{\phi}_{+}\overline{\phi}_{-}\sim\xi^{4}\,,$
(4.63)

which satisfies the above F-flatness conditions to the order considered here
(up to $N=6$), at it effectively diminishes the associated contributions in
Eqs. (4.58)-(4.60) down to $N=7$ order.

Having solved the flatness conditions, let us now consider higher order
contributions to the singlet masses and mixings. More particularly, we focus
first on bilinear couplings comprising the inflaton field $\phi_{0}$. Using
eq. (4.48), these correspond to interactions of the form
$\left(\phi_{3}\varphi\right)\varphi_{1}\dots\varphi_{N-2}$ and
$\left(\overline{\phi}_{3}\varphi\right)\varphi_{1}\dots\varphi_{N-2}$, where
$\varphi\in\left\\{\Phi_{I},\Phi_{a},\overline{\Phi}_{a},\phi_{i},\overline{\phi}_{i}\right\\},I=1,2,4,a=13,23,i=1,2,3,4,45$
and $\varphi_{\alpha}$ stand for singlet or hidden sector field VEVs. Taking
into account our F/D-flatness solution we find a single non-vanishing such
coupling up to and including $N=6$ order NR terms in the superpotential

$\displaystyle\left[\overline{\phi}_{3}\Phi_{4}\right]\left(D_{1}{\cdot}D_{5}\right)\left(T_{1}{\cdot}T_{4}\right)\,.$
(4.64)

For completeness, we also check all bilinear couplings of the same type
involving $\Phi_{4}$. We find a single mixing term at $N=6$:

$\displaystyle\left[\Phi_{4}\Phi_{12}\right]\left(D_{1}{\cdot}D_{5}\right)^{2}\,.$
(4.65)

This coupling does not play an important rôle in our analysis as it simply
introduces a small mixing of $\Phi_{4}$ with $\Phi_{12}$ that became
superheavy at tree-level. Following the discussion in Section 3, we are led to
identify $\Phi_{4}$ with the goldstino superfield.

Next, we compute all trilinear couplings induced by NR terms of the form
$\left(\phi_{3}^{2}\Phi_{4}\right)\varphi_{1}\dots\varphi_{N-3}$,
$\left(\overline{\phi}_{3}^{2}\Phi_{4}\right)\varphi_{1}\dots\varphi_{N-3}$
and
$\left(\phi_{3}\overline{\phi}_{3}\Phi_{4}\right)\varphi_{1}\dots\varphi_{N-3}$.
It turns out that up to and including $N=7$ no such coupling exists. At $N=8$,
we find a unique term

$\displaystyle\left[\overline{\phi}_{3}^{2}\Phi_{4}\right]\left(D_{1}{\cdot}D_{4}\right)\left(T_{1}{\cdot}T_{4}\right)\Phi_{31}\,.$
(4.66)

Altogether, the inflaton related NR effective superpotential reads

$\displaystyle W_{I}$
$\displaystyle=g_{s}C_{6}\left(g_{s}\sqrt{2\alpha^{\prime}}\right)^{3}\,\overline{\phi}_{3}\Phi_{4}\langle{D_{1}}\rangle{\cdot}\langle{D_{5}}\rangle\,\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle$
(4.67) $\displaystyle\hskip
22.76228pt+g_{s}C_{8}\left(g_{s}\sqrt{2\alpha^{\prime}}\right)^{5}\,\overline{\phi}_{3}^{2}\Phi_{4}\langle{D_{1}}\rangle{\cdot}\langle{D_{4}}\rangle\,\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle\langle{\Phi_{31}}\rangle\,,$

where $C_{6},C_{8}$ stand for the numerical values of the correlators
associated to the $N=6$ and $N=8$ NR couplings, respectively and we restored
the mass units.

In order to make an estimate of the resulting inflaton superpotential, we
introduce a typical VEV $\langle{\phi}\rangle$ as dummy variable and rewrite
(4.67), using also (4.50), as:

$\displaystyle
W_{I}=\zeta^{4}\Phi_{4}\left(\frac{\gamma}{g_{s}\sqrt{2\alpha^{\prime}}}\phi_{0}+\delta\zeta\phi_{0}^{2}\right)\,,$
(4.68)

where $\gamma,\delta$ and $\zeta$ are dimensionless parameters given by:

$\displaystyle\gamma$
$\displaystyle=-g_{s}C_{6}\frac{\langle{D_{1}}\rangle{\cdot}\langle{D_{5}}\rangle\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle}{\langle\phi\rangle^{4}}\cos\omega\,,$
(4.69) $\displaystyle\delta$
$\displaystyle=\hphantom{+}g_{s}C_{8}\frac{\langle{D_{1}}\rangle{\cdot}\langle{D_{4}}\rangle\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle\langle\Phi_{31}\rangle}{\langle\phi\rangle^{5}}\cos^{2}\omega\,,$
(4.70) $\displaystyle\zeta$ $\displaystyle={\langle\phi\rangle
g_{s}\sqrt{2\alpha^{\prime}}}\,.$ (4.71)

A typical VEV $\langle{\phi}\rangle$ satisfying the D-term conditions
(4.4)-(4.8) is of order $\xi$, leading to an estimate:

$\displaystyle\zeta\sim{\xi
g_{s}\sqrt{2\alpha^{\prime}}}=\frac{g_{s}}{2\pi}\simeq 0.08\,,$ (4.72)

where we used $g_{s}=g/\sqrt{2}$, with $g$ the GUT gauge coupling,
corresponding to the standard convention 1/2 for the trace of the square of
gauge group generators in the fundamental representation.

At this point, one can easily reconstruct the Starobinsky-like inflationary
potential of Section 3 from the superpotential (4.68), by renaming
$\phi_{0}=y,\Phi_{4}=z$:

$\displaystyle W_{I}=M_{I}z\left(y-\lambda y^{2}\right)\,,$ (4.73)

where

$M_{I}=\zeta^{4}{\gamma\over
g_{s}}{1\over\sqrt{2\alpha^{\prime}}}\simeq\zeta^{4}C_{6}\cos\omega{1\over\sqrt{2\alpha^{\prime}}}$
(4.74)

is the inflation scale and

$\lambda=-g_{s}\zeta{\delta\over\gamma}\sqrt{2\alpha^{\prime}}M_{P}\simeq
g_{s}\zeta{C_{8}\over
C_{6}}{\langle{D_{1}}\rangle{\cdot}\langle{D_{4}}\rangle\over\langle{D_{1}}\rangle{\cdot}\langle{D_{5}}\rangle}\sqrt{2\alpha^{\prime}}M_{P}\,.$
(4.75)

Then, for $\gamma\sim\zeta\delta$, obtained for instance by choosing

$\displaystyle\omega\sim 0\ ,\
\langle{D_{1}}\rangle{\cdot}\langle{D_{5}}\rangle\sim\zeta\langle\phi\rangle^{2}\
,\
\langle{D_{1}}\rangle{\cdot}\langle{D_{4}}\rangle\sim\langle\phi\rangle^{2}\
,\
\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle\sim\langle\phi\rangle^{2}\
,\ \langle\Phi_{31}\rangle\sim\langle\phi\rangle\,,$ (4.76)

and using the relation between the reduced Planck mass $M_{P}=1/\sqrt{8\pi G}$
and the heterotic string scale $M_{s}\equiv 1/\sqrt{2\alpha^{\prime}}$

$M_{P}^{2}={2\over
g_{s}^{2}\alpha^{\prime}}\quad\Rightarrow\quad{1\over\sqrt{2\alpha^{\prime}}}\simeq
5\times 10^{17}\,{\rm GeV}\,,$ (4.77)

we obtain $M_{I}\simeq\zeta^{5}C_{6}M_{s}\sim 10^{13}$ GeV and $\lambda\simeq
g_{s}M_{P}/M_{s}$ an order one tuneable parameter. It is curious to notice
that by tuning $\lambda$ to unity, one can produce in the string spectrum a
scalar that has exactly the properties of an $R^{2}$ term in the effective
action.

Thus, the effective superpotential $W_{I}$ is in full agreement with (3.15)
for a range of VEVs of the singlet and hidden sector fields in the theory
compatible with F/D-flatness constraints. Putting everything together we
conclude that the Starobinsky-like inflationary scenario is fully realisable
in the string derived “revamped” flipped ${SU(5)}\times{U(1)}$ model.

Note that the $\lambda_{6}$ term $F_{4}{\bar{F}}_{5}\phi_{3}$ in eq. (2.1)
metamorphoses, using eq. (4.50), to $\sin\omega F_{4}{\bar{F}}_{5}\phi_{0}$,
involving the inflaton field $\phi_{0}$ thus the $\lambda_{6}$ coupling
becomes $\sin\omega/\sqrt{2}$. This is a highly desirable result, because as
has been found in Refs. [20, 22] $\lambda_{6}$ should be $\mathrel{\lower
2.5pt\vbox{\hbox{$<$}\hbox{$\sim$}}}10^{-3}$ in order to provide appropriate
reheating, baryogenesis and light neutrino masses. Thus
$\langle\phi_{4}\rangle/\langle{\bar{\phi}}_{4}\rangle$ should be $<<1\,$,
compatible with the F- and D-flat directions imposed constraints on the gauge
singlets VEVs. Indeed, the following solution to the flatness conditions
satisfy this requirement:

$\displaystyle\overline{\phi}_{4}\sim\xi^{2}\ ,\ \phi_{-}\sim\xi^{3}\ ,\
\overline{\phi}_{+}\sim\xi^{3}\ ,\ $ (4.78)

and $\phi_{4}/\overline{\phi}_{4}\sim 10^{-3}$. This is compatible with (4.62)
(due to the presence of the NR term) and (4.63), provided
$\overline{\phi}_{45}\sim\xi^{3}$ in order to satisfy (4.44). After applying
(4.40), (4.53) and (4.54)-(4.60) the D-flatness equations reduce to:

$\displaystyle\left|\phi_{45}\right|^{2}-\frac{\left|T_{3}\right|^{2}}{2}\sim\xi^{2}\,,$
(4.79)
$\displaystyle\left|\overline{\phi}_{-}\right|^{2}+\left|{\phi}_{+}\right|^{2}-\frac{\left|T_{3}\right|^{2}}{2}\sim\xi^{2}\,,$
(4.80)
$\displaystyle\left|\Phi_{31}\right|^{2}-\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{4}\right|^{2}-\left|D_{5}\right|^{2}\right)-\frac{1}{2}\left(\left|T_{1}\right|^{2}+\left|T_{3}\right|^{2}-\left|T_{4}\right|^{2}\right)=3\xi^{2}\,,$
(4.81) $\displaystyle
2\left|\phi_{+}\right|^{2}+\left|D_{1}\right|^{2}+\left|D_{4}\right|^{2}+\left|T_{1}\right|^{2}-\left|T_{4}\right|^{2}\sim
0\,,$ (4.82)

which can be easily satisfied.

##  5 Conclusions and outook

In summary, in this work, we have investigated the cosmology of a string
derived flipped $SU(5)\times U(1)$ model in the free-fermionic framework of 4d
heterotic superstrings. We have shown that a successful Starobinsky-type
inflation can be accommodated with the inflaton being the superpartner of a
state that mixes with a right-handed neutrino (linear combination) and an
appropriate superpotential involving also the goldstino superfield. The
effective field theory is a $N=1$ no-scale supergravity with an exactly
calculable (to all orders in $\alpha^{\prime}$) Kähler potential and a
superpotential induced perturbatively around the fermionic point via a set of
VEVs for $SU(5)\times U(1)$ singlet fields, driven by an anomalous $U(1)_{A}$
gauge symmetry generally present in chiral heterotic string constructions

An important point is that during inflation, the GUT symmetry group
$SU(5)\times U(1)$ remains unbroken; its breaking occurs via a first order
phase transition at lower energies, after the end of inflation. This is a
desirable feature, since the present analysis of the inflationary period does
not a priori affect the $SU(5)\times U(1)$ breaking vacuum and the resulting
particle phenomenology that has been studied extensively in the past. Another
related feature is that soft supersymmetry breaking terms do not play any rôle
either during inflation, with the inflaton rolling down to a supersymmetric
minimum before the phase transition to the low energy vacuum occurs.

In this paper, we performed a detailed analysis of the non-renormalisable
terms in the superpotential, up to dimension-eight, and identified a set of
$SU(5)\times U(1)$ singlet VEVs consistent with the F- and D-flatness
conditions, that generate the requested inflationary superpotential for the
inflaton and goldstino superfields; the latter were appropriately chosen among
the gauge singlets of the model with the desired Kähler potential, so that the
resulting scalar potential leads to a Starobinsky-type inflation consistent
with the CMB observations.

Several open questions remain to be studied. In particular, a stabilisation
mechanism for the dilaton is desirable in heterotic string models using either
non-perturbative effects and/or magnetic and 3-form fluxes in analogy with
type II models. Obviously, our result is a proof of concept showing the
existence of a general class of models sharing similar properties. On the
other hand, it is astonishing that the explicit revamped flipped $SU(5)$
construction, built 30 years ago, accommodates remarkably well a low energy
phenomenology both in particle physics and cosmology. A detailed
phenomenological analysis combining all aspects is currently under
investigation.

## Acknowledgements

J.R. would like to thank LPTHE for hospitality and financial support. Work
supported in part by a CNRS-PICS grant 07964. The work of DVN was supported in
part by the DOE grant DE-FG02-13ER42020 at Texas A&M University and in part by
the Alexander S. Onassis Public Benefit Foundation. We would like to thank
Natsumi Nagata and especially Keith Olive for enlightening discussions.

##  Appendix A String construction and spectrum of the flipped
$SU(5)\times{U(1)}$ model

The revamped flipped $SU(5)\times{U(1)}$ string model is constructed using the
free fermionic formulation of the heterotic superstring. In this framework a
string model is defined by a set of basis vectors
$B=\left\\{\beta_{1},\beta_{2},\dots,\beta_{n}\right\\}$ and a set of phases
$C_{ij}=c\left[\beta_{i}\atop\beta_{j}\right]\,,i,j=1,\dots,n$. The former is
associated with the transformation properties of the world-sheet fermions
while the later is related to generalised GSO projections. The revamped
flipped model is generated using 8 basis vectors, namely

$\displaystyle\beta_{1}=S$
$\displaystyle=\left\\{\psi^{\mu},\chi^{1,\dots,6}\right\\}\,,$
$\displaystyle\beta_{2}=b_{1}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{12},y^{3,\dots,6};\overline{y}^{3,\dots,6},\overline{\psi}^{1,\dots,5},\overline{\eta}^{1}\right\\}\,,$
$\displaystyle\beta_{3}=b_{2}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{34},y^{12},\omega^{56};\overline{y}^{12},\overline{\omega}^{56},\overline{\psi}^{1,\dots,5}\overline{\eta}^{2}\right\\}\,,$
$\displaystyle\beta_{4}=b_{3}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{56},\omega^{1,\dots,4};\overline{\omega}^{1,\dots,4},\overline{\psi}^{1,\dots,5},\overline{\eta}^{3}\right\\}\,,$
(A.1) $\displaystyle\beta_{5}=b_{4}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{12},y^{36},\omega^{45};\overline{y}^{36},\overline{\omega}^{45},\overline{\psi}^{1,\dots,5},\overline{\eta}^{1}\right\\}\,,$
$\displaystyle\beta_{6}=b_{5}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{34},y^{26},\omega^{15};\overline{y}^{26},\overline{\omega}^{15},\overline{\psi}^{1,\dots,5},\overline{\eta}^{2}\right\\}\,,$
$\displaystyle\beta_{7}=\zeta$
$\displaystyle=\left\\{\overline{\phi}^{1,\dots,8}\right\\}\,,$
$\displaystyle\beta_{8}=\alpha$
$\displaystyle=\Bigl{\\{}y^{46},\omega^{46};\overline{y}^{46},\overline{\omega}^{2346},\underbrace{\overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,4}}_{\frac{1}{2},\dots,\frac{1}{2}},\overline{\phi}^{56}\Bigr{\\}}\,,$

where included fermions are periodic unless indicated otherwise (i.e.
$\frac{1}{2}$ stands for fermions twisted by $-i$). The associated GGSO phase
matrix elements are
$C_{ij}=c\left[\beta_{i}\atop\beta_{j}\right]=\exp(i\pi\tilde{c}_{ij})$, where

$\displaystyle\tilde{c}=\bordermatrix{~{}&S&b_{1}&b_{2}&b_{3}&b_{4}&b_{5}&\zeta&\alpha\cr
S&0&0&0&0&0&0&1&1\cr b_{1}&1&1&1&1&1&1&1&-1/2\cr b_{2}&1&1&1&1&1&1&1&-1/2\cr
b_{3}&1&1&1&1&1&1&1&1\cr b_{4}&1&1&1&1&1&1&1&-1/2\cr
b_{5}&1&1&1&1&1&1&1&+1/2\cr\zeta&1&1&1&1&1&1&1&1\cr\alpha&1&1&1&1&1&0&1&-1/2\cr}\,.$
(A.2)

The above set of basis vectors and GGSO phases define a $N=1$ supersymmetric
string model exhibiting
$SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ gauge symmetry.
We will consider the flipped $SU(5)\times{U(1)}$ group factor as the
“observable” gauge group and the ${SU(4)}\times{SO(10)}$ group factor as the
“hidden” gauge group. The four additional abelian factors
${U(1)}^{4}=\prod_{i=1}^{4}{U(1)}_{i}$ corresponding to the world-sheet
currents
$\overline{\eta}^{1}\overline{\eta}^{1\ast},\overline{\eta}^{2}\overline{\eta}^{1\ast},\overline{\eta}^{3}\overline{\eta}^{3\ast},\overline{\omega}^{2}\overline{\omega}^{3}$
appear to be anomalous; ${\rm Tr}{U(1)}_{1}=-36,{\rm Tr}{U(1)}_{2}=-12,{\rm
Tr}{U(1)}_{3}=-24,{\rm Tr}{U(1)}_{4}=-12$. Redefining appropriately we obtain
three $U(1)$ combinations free of gauge and mixed gravitational anomalies

$\displaystyle{U(1)}^{\prime}_{1}$ $\displaystyle={U(1)}_{3}+2U(1)_{4}\,,$
(A.3) $\displaystyle{U(1)}^{\prime}_{2}$
$\displaystyle={U(1)}_{1}-3U(1)_{2}\,,$ (A.4)
$\displaystyle{U(1)}^{\prime}_{3}$
$\displaystyle=3{U(1)}_{1}+{U(1)}_{2}+4{U(1)}_{3}-2{U(1)}_{4}\,,$ (A.5)

and one anomalous abelian group factor

$\displaystyle{U(1)}_{A}=-3{U(1)}_{1}-{U(1)}_{2}+2{U(1)}_{3}-{U(1)}_{4}\ \,,\
{\rm Tr}{U(1)}_{A}=180\,.$ (A.6)

The massless matter spectrum comprises: (i) “Observable” sector matter
particles, that is, states charged solely under the
$SU(5)\times{U(1)}\times{U(1)}^{4}$ group factor, listed in Table 1. These
include three chiral fermion families as well as a family/anti-family pair,
accommodated in $SO(10)$ spinorials/anti-spinorials, arising from the sectors
$b_{1},b_{2},b_{3},b_{4},b_{5}$, and three fiveplet/anti-fiveplet pairs,
accommodated in $SO(10)$ vectorials, from the sectors $S,S+b_{4}+b_{5}$.
Bosonic partners come from the sectors $S+b_{i},i=1,\dots,4$ and
$0,b_{4}+b_{5}$ respectively. (ii) “Hidden” sector matter particles, that is,
states exclusively charged under ${SU(4)}\times{SO(10)}\times{U(1)}^{4}$
listed in Table 2. These arise from the sectors
$(S)+b_{i}+2\alpha(+\zeta),i=1,\dots,4$. (iii) Exotic fractionally charged
states coming from the sectors $(S)+b_{1}+\alpha(+\zeta)$,
$(S)+b_{1}+b_{4}+b_{5}+\alpha(+\zeta)$,
$(S)+b_{2}+b_{3}+b_{5}+\alpha(+\zeta)$,
$(S)+b_{1}+b_{2}+b_{4}+\alpha(+\zeta)$. $(S)+b_{2}+b_{4}+\alpha(+\zeta)$,
$(S)+b_{4}+\alpha(+\zeta)$. These are listed in Table 3.

| $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$S$
$h_{1}$ | ${\mathbf{5}}$ | $-1$ | $+1$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{1}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $-1$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$h_{2}$ | ${\mathbf{5}}$ | $-1$ | $0$ | $+1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{2}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $0$ | $-1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$h_{3}$ | ${\mathbf{5}}$ | $-1$ | $0$ | $0$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{3}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $0$ | $0$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{12}$ | ${\mathbf{1}}$ | $0$ | $+1$ | $-1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{12}$ | ${\mathbf{1}}$ | $0$ | $-1$ | $+1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\Phi}_{13}$ | ${\mathbf{1}}$ | $0$ | $+1$ | $0$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{13}$ | ${\mathbf{1}}$ | $0$ | $-1$ | $0$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{23}$ | ${\mathbf{1}}$ | $0$ | $0$ | $+1$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{23}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-1$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{I}\,,I=1,\dots,5$ | ${\mathbf{1}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{1}$
$F_{1}$ | ${\mathbf{10}}$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{1}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\ell}^{c}_{1}$ | ${\mathbf{1}}$ | $+{5/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{2}$
$F_{2}$ | ${\mathbf{10}}$ | $+{1/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{2}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\ell}^{c}_{2}$ | ${\mathbf{1}}$ | $+{5/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{3}$
$F_{3}$ | ${\mathbf{10}}$ | $+{1/2}$ | $0$ | $0$ | $+{1/2}$ | $-{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{3}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $0$ | $+{1/2}$ | $+{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\ell}^{c}_{3}$ | ${\mathbf{1}}$ | $+{5/2}$ | $0$ | $0$ | $+{1/2}$ | $+{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{4}$
$F_{4}$ | ${\mathbf{10}}$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${f}_{4}$ | ${{\mathbf{5}}}$ | $+{3/2}$ | $+{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\ell}^{c}_{4}$ | ${\mathbf{1}}$ | $+{5/2}$ | $+{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{5}$
$\overline{F}_{5}$ | $\overline{\mathbf{10}}$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{5}$ | ${{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\ell}^{c}_{5}$ | ${\mathbf{1}}$ | $-{5/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$S+b_{4}+b_{5}$
$h_{45}$ | ${\mathbf{5}}$ | $-1$ | $-{1/2}$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{h}_{45}$ | $\overline{\mathbf{5}}$ | $+1$ | $+{1/2}$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{45}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $+{1/2}$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{45}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $-{1/2}$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{+}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $+1$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{+}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $-1$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{-}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $-1$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{-}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $+1$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{i}\,,i=1,\dots,4$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{i}\,,i=1,\dots,4$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
Table 1: “Observable” sector massless matter states and their $SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum numbers. | $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$b_{1}+2\alpha\,\left(+\zeta\right)$
$D_{1}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{1}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{2}+2\alpha\,\left(+\zeta\right)$
$D_{2}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{2}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{3}+2\alpha\,\left(+\zeta\right)$
$D_{3}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $-{1/2}$ | $0$ | $+{1/2}$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{3}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $-{1/2}$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{4}+2\alpha\,\left(+\zeta\right)$
$D_{4}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{4}$ | ${\mathbf{1}}$ | $0$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{5}+2\alpha\,\left(+\zeta\right)$
$D_{5}$ | ${\mathbf{1}}$ | $0$ | $+{1/2}$ | $0$ | $-{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{5}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
Table 2: “Hidden sector” massless matter states and their $SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum numbers. | $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$b_{1}\pm\alpha\,\left(+\zeta\right)$
$\overline{X}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $+{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$\overline{X}_{2}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{1}+b_{4}+b_{5}\pm\alpha\,\left(+\zeta\right)$
${Y}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
${Y}_{2}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$b_{2}+b_{3}+b_{5}\pm\alpha\,\left(+\zeta\right)$
${Z}_{2}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{3/4}$ | $+{1/4}$ | $0$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Z}_{2}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{3/4}$ | $+{1/4}$ | $-{1/4}$ | $0$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{1}+b_{2}+b_{4}\pm\alpha\,\left(+\zeta\right)$
${Y}_{2}^{\prime}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Y}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$S+b_{2}+b_{4}\pm\alpha\,\left(+\zeta\right)$
${Z}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Z}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{4}\pm\alpha\,\left(+\zeta\right)$
${X}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{X}_{2}^{\prime}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
Table 3: Exotic fractionally charged massless matter states and their
$SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum
numbers.

##  Appendix B Calculation of the potential in no-scale supergravity

Using the following Kähler potential and superpotential

$\displaystyle K$
$\displaystyle=-2\ln\left(1-\frac{1}{2}|z|^{2}\right)-2\ln\left(1-\left|y\right|^{2}\right)\,,$
(B.1) $\displaystyle W$ $\displaystyle=\mu z\left(y-\lambda y^{2}\right)\,,$
(B.2)

we get

$\displaystyle K_{z}$
$\displaystyle=\frac{\bar{z}}{1-\frac{\left|z\right|^{2}}{2}}\,,$ (B.3)
$\displaystyle K_{y}$ $\displaystyle=\frac{2\bar{y}}{1-\left|y\right|^{2}}\,,$
(B.4) $\displaystyle K_{z\bar{z}}$
$\displaystyle=\frac{1}{\left(1-\frac{\left|z\right|^{2}}{2}\right)^{2}}\,,$
(B.5) $\displaystyle K_{y\bar{y}}$
$\displaystyle=\frac{2}{\left(1-\left|y\right|^{2}\right)^{2}}\,,$ (B.6)

and

$\displaystyle D_{z}W$ $\displaystyle=\mu
y\frac{\left(2+\left|z\right|^{2}\right)\left(1-\lambda
y\right)}{2-\left|z\right|^{2}}\,,$ (B.7) $\displaystyle D_{y}W$
$\displaystyle=\mu z\frac{1-2\lambda
y+\left|y\right|^{2}}{1-\left|y\right|^{2}}\,.$ (B.8)

The potential reads

$\displaystyle V$
$\displaystyle=e^{K}\left(\left|D_{i}W\right|^{2}-3\left|W\right|^{2}\right)=\frac{\mu^{2}P(y,z)}{\left(1-\left|y\right|^{2}\right)^{2}\left(2-\left|z\right|^{2}\right)^{2}}\,,$
(B.9)

where

$\displaystyle P$
$\displaystyle\\!\\!=\\!\\!\left\\{4|y|^{2}\left(1\\!-\\!2\lambda|y|\cos\theta\\!+\\!\lambda^{2}|y|^{2}\right)+2\left(1\\!-\\!\left|y\right|^{2}\right)\left(1\\!-\\!|y|^{2}\\!+\\!4\lambda^{2}\left|y\right|^{2}\\!-\\!4\lambda\left|y\right|\cos\theta\right)\left|z\right|^{2}\right.$
$\displaystyle\left.\ \ \
+\left|y\right|^{2}\left(1-2\lambda|y|\cos\theta+\lambda^{2}|y|^{2}\right)\left|z\right|^{4}\right\\}\,.$
(B.10)

Minimizing with respect to $\theta$ we obtain potential minima for

$\displaystyle\cos\theta=\text{sgn}(\lambda)\,.$ (B.11)

In the region $\left|y\right|\sim 1$ the potential diverges unless the
numerator is proportional to $\left(1-\left|y\right|\right)$. The remainder of
the division of the numerator $P(y,z)$ over $\left(1-\left|y\right|\right)$ is

$\displaystyle\mu^{2}(4+\left|z\right|^{4})\left|\left(1-2\lambda\cos\theta+\lambda^{2}\right)\right|_{\cos\theta=\text{sgn}\left(\lambda\right)}=\mu^{2}(4+\left|z\right|^{4})1\left(1-|\lambda|\right)^{2}\,.$
(B.12)

This vanishes for $|\lambda|=1$ and gives

$\displaystyle
V=\frac{\mu^{2}\left[4|y|^{2}+(2-4|y|-6|y|^{2})\left|z\right|^{2}+|y|^{2}\left|z\right|^{4}\right]}{(1+\left|y)\right|^{2}\left(2-\left|z\right|^{2}\right)^{2}}\,.$
(B.13)

Expanding the potential around $z=0$ we get

$\displaystyle
V=\frac{\mu^{2}|y|^{2}}{(1+|y|)^{2}}+\mu^{2}\frac{(1-|y|)^{2}-2|y|^{2}}{2(1+|y|)^{2}}\left|z\right|^{2}+\dots\,.$
(B.14)

In case we take into account the dilaton the minimum remains at
$\cos\theta=\text{sgn}(\lambda)$ and the elimination of the pole at
$\left|y\right|=1$ requires again $|\lambda|=1$. Using this the potential
becomes

$\displaystyle V$
$\displaystyle\\!\\!=\\!\\!\left.e^{K}\left(\left|D_{i}W\right|^{2}\\!-\\!2\left|W\right|^{2}\right)\right|_{\cos\theta=\text{sgn}(\lambda),|\lambda|=1}\\!\\!=\\!\mu^{2}\frac{4|y|^{2}+(2-4|y|-2|y|^{2})\left|z\right|^{2}+|y|^{2}\left|z\right|^{4}}{(S+{\bar{S}})\left(1+|y|\right)^{2}\left(2-\left|z\right|^{2}\right)^{2}}$
$\displaystyle=\frac{\hat{\mu}^{2}}{(1+|y|)^{2}}\left(|y|^{2}+\frac{1}{2}\frac{(1-|y|)^{2}\left|z\right|^{2}}{\left(1-\frac{\left|z\right|^{2}}{2}\right)^{2}}\right)=\frac{\hat{\mu}^{2}|y|^{2}}{(1+|y|)^{2}}+\frac{\hat{\mu}^{2}(1-|y|)^{2}}{(1+|y|)^{2}}\left|z\right|^{2}+\dots\,,$
(B.15)

where we defined $\hat{\mu}^{2}\equiv\mu^{2}/(S+{\bar{S}})$.

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