# From Minkowski to de Sitter Vacua with Various Geometries

Constantinos Pallis
Laboratory of Physics, Faculty of Engineering, Aristotle University of
Thessaloniki, GR-541 24 Thessaloniki, GREECE
e-mail address<EMAIL_ADDRESS>
###### Abstract

Abstract: New no-scale supergravity models with F-term SUSY breaking are
introduced, adopting Kähler potentials parameterizing flat or curved (compact
or non-compact) Kähler manifolds. We systematically derive the form of the
superpotentials leading to Minkowski vacua. Combining two types of these
superpotentials we can also determine de Sitter or anti-de Sitter vacua. The
construction can be easily extended to multi-modular settings of mixed
geometry. The corresponding soft SUSY-breaking parameters are also derived.

PACs numbers: 12.60.Jv, 04.65.+e

###### Contents

  1. I Introduction
  2. II Start-Up Considerations
  3. III Flat Moduli Geometry
    1. A Uni-Modular Model
    2. B Multi-Modular Model
  4. IV Curved Moduli Geometry
    1. A Uni-Modular Model
    2. B Multi-Modular Model
  5. V Generalization
  6. VI Link to the Observable Sector
  7. VII Conclusions
  8. A Mass Formulae In SUGRA
  9. B Half-Plane Parametrization

##  I Introduction

Within _Supergravity_ (SUGRA) nilles ; mass , breaking _Supersymmetry_ (SUSY)
on a sufficiently flat background requires a huge amount of fine tuning,
already at the classical level – see e.g. Ref. polonyi ; hall . Besides
remarkable exceptions presented recently susyr , the so-called no-scale models
noscale ; old ; noscale18 ; noscale19 ; Rgauged ; nsreview ; fabio ; burgess ;
roest  provide an elegant framework which alleviates the problem above since
SUSY is broken with naturally vanishing vacuum energy along a flat direction.
On the other hand, the discovery of the accelerate expansion of the present
universe plcp  motivates us to develop models with _de Sitter_ (dS) – or even
_anti-dS_ (AdS) – vacua which may explain this expansion – independently of
the controversy vafa ; lindev ; sevilla ; ferara  surrounding this kind of
(meta) stable vacua within string theory.

In two recent papers noscale18 ; noscale19 , a systematic derivation of dS/AdS
vacua is presented in the context of the no-scale SUGRA without invoking any
external mechanism of vacuum uplifting such as through the addition of anti-D3
brane contributions kallosh  or extra Fayet-Iliopoulos terms antst . Namely,
these vacua are achieved by combining two distinct Minkowski vacua taking as
initial point the Kähler potential parameterizing the non-compact
$SU(1,1)/U(1)$ Kähler manifold in half-plane coordinates, $T$ and $T^{*}$.
Possible instabilities along the imaginary direction of the $T$ field can be
cured by introducing mild deformations of the adopted geometry. The analysis
has been extended to incorporate more than one superfields in conjunction with
the implementation of observationally successful inflation noscaleinfl ;
nsreview .

In this paper we show that the method above has a much wider applicability
since it remains operational for flat spaces or curved ones. This is possible
since the no-scale “character” of the models, as defined above, stems from the
existence of a flat direction with SUSY broken along it, and not from the
adopted moduli geometry. We parameterize the curved spaces of our models with
the Poincaré disk coordinates $Z_{\alpha}$ and $Z^{*}_{\bar{\alpha}}$ which,
although are widely adopted within the inflationary model building linde ;
alinde ; sor , they are not frequently employed for establishing SUSY-breaking
models – cf. Ref. susyr ; Rgauged . This parametrization gives us the
opportunity to go beyond the non-compact geometry noscale18 ; noscale19  and
establish SUSY-breaking scenaria with compact su11  or “mixed” geometry. In
total, we here establish three novel uni-modular no-scale models and discuss
their extensions to the multi-modular level. In all cases, we show that a
subdominant quartic term noscale18 ; susyr  in the Kähler potential stabilizes
the sgoldstino field to a specific vacuum and provides mass for its scalar
component without disturbing, though, the constant vacuum energy density. This
can be identified with the present cosmological constant by finely tuning one
parameter of the model whereas the others can be adjusted to perfectly natural
values. If we connect, finally, our hidden sectors with some sample observable
ones, non-vanishing _soft SUSY-breaking_ (SSB) parameters soft , of the order
of the gravitino mass can be readily determined at the tree level.

We start our presentation with a simplified generic argument which outlines
the transition from Minkowski to dS/AdS vacua in Sec. II. We then detail our
models adopting first – in Sec. III – flat moduli geometry and then – see Sec.
IV – two versions of curved geometry. Generalization of our findings
displaying multi-modular models with mixed geometry is presented in Sec. V. We
also study in Sec. VI the communication of the SUSY breaking to the observable
sector by computing the SSB terms. We summarize our results in Sec. VII. Some
useful formulae related to derivation of mass spectra in SUGRA with dS/AdS
vacua is arranged in Appendix A. In Appendix B we show the consistency of our
results with those in Ref. noscale18 ; noscale19  translating the first ones
in the language of the $T-T^{*}$ coordinates.

Unless otherwise stated, we use units where the reduced Planck scale $m_{\rm
P}=2.4\cdot 10^{18}~{}{\mbox{\rm GeV}}$ is taken to be unity and the star (∗)
denotes throughout complex conjugation. Also, no summation convention is
applied over the repeated Latin indices $\ell,i$ and $j$.

##  II Start-Up Considerations

The generation mechanism of dS/AdS vacua from a pair of Minkowski ones can be
roughly established, if we consider a uni-modular model without specific
geometry. In particular, we adopt a Kähler potential $K=K(Z,Z^{*})$ and
attempt to determine an expression for the superpotential $W=W(Z)$ so as to
construct a no-scale scenario.

The SUGRA potential $V$ based on $K$ and $W$ from Eq. (98) is written as

$V=e^{K}\left(g_{K}^{-1}\left|\partial_{Z}W+WK_{Z}\right|^{2}-3|W|^{2}\right),$
(1)

where $g_{K}^{-1}=K_{ZZ^{*}}^{-1}=K^{ZZ^{*}}$. Suppose that there is an
expression $W=W_{0}(Z)$ which assures that the direction $Z=Z^{*}$ is
classically flat with $V=0$, i.e., it provides a continuum of Minkowski vacua.
The determination of $W_{0}$, based on Eq. (1), entails

$g_{K}^{-1}\left(W_{0}^{\prime}+W_{0}K_{Z}\right)^{2}=3W_{0}^{2}~{}\Rightarrow~{}\frac{dW_{0}}{dZW_{0}}=\pm\sqrt{3g_{K}}-K_{Z}$
(2)

with Eq. (99) being satisfied – the relevant conditions may constrain the
model parameters once $K$ is specified. Here prime stands for derivation _with
respect to_ (w.r.t) $Z$. Eq. (2) admits obligatorily two solutions

$W_{0}^{\pm}=m\exp\left(\pm\int dZ\sqrt{3g_{K}}-\int dZK_{Z}\right)$ (3)

with $m$ a mass parameter. For the $K$’s considered below, it is easy to
verify that

$\int dZK_{Z}=K/2$ (4)

up to a constant of integration. E.g., if $K=|Z|^{2}$, then $K_{Z}=Z$ for
$Z^{*}=Z$ and $\int dZK_{Z}=Z^{2}/2=K/2$.

According to Ref. noscale18 ; noscale19 , the appearance of dS/AdS vacua is
attained, if we consider the following linear combination of $W_{0}^{\pm}$ in
Eq. (3)

$W_{\Lambda}=C^{+}W_{0}^{+}-C^{-}W_{0}^{-},$ (5)

where $C^{-}$ and $C^{+}$ are non-zero constants. As can be easily checked,
$W_{\Lambda}$ does not consist solution of Eq. (2). It offers, however, the
achievement of a technically natural dS/AdS vacuum since its substitution into
Eq. (98) yields

$\displaystyle V_{\Lambda}$
$\displaystyle=e^{K}\left(g_{K}^{-1}\left(W_{\Lambda}^{\prime}+W_{\Lambda}K_{Z}\right)^{2}-3W_{\Lambda}^{2}\right)$
(6) $\displaystyle=12e^{K}C^{-}C^{+}W_{0}^{-}W_{0}^{+}=12m^{2}C^{-}C^{+},$

where we take into account Eqs. (3) and (4). Rigorous validation and extension
(to more superfields) of this method can be accomplished via its application
to specific working models. This is done in the following sections.

Let us, finally, note that $V_{\Lambda}$ can be identified with the present
cosmological constant by demanding

$V_{\Lambda}=\Omega_{\Lambda}\rho_{\rm c0}=7.2\cdot 10^{-121}m_{\rm P}^{4},$
(7)

where $\Omega_{\Lambda}=0.6889$ and $\rho_{\rm c0}=2.4\cdot
10^{-120}h^{2}m_{\rm P}^{4}$ with $h=0.6732$ plcp  is the density parameter of
dark energy and the current critical energy density of the universe.

##  III Flat Moduli Geometry

We focus first on the models with flat internal geometry and describe below
their version for one – see Sec. A – or more – see Sec. B – moduli.

###  A Uni-Modular Model

Our initial point is the Kähler potential

$K_{\rm f}=|Z|^{2}-k^{2}Z_{\rm v}^{4}$ (8)

where we include the stabilization term

$Z_{\rm v}=Z_{+}-\sqrt{2}{\rm v}~{}~{}\mbox{with}~{}~{}Z_{\pm}=Z\pm Z^{*}.$
(9)

Here $k$ and ${\rm v}$ are two real free parameters. Small $k$ values are
completely natural, according to the ’t Hooft argument symm , since $K_{\rm
f}$ enjoys an enhanced $U(1)$ symmetry which is exact for $k=0$. It is evident
that the $Z$ space defined by $K_{\rm f}$ is flat with metric
$\left\langle{K_{ZZ^{*}}}\right\rangle=1$ along the stable configurations

$\displaystyle Z=Z^{*}$ $\displaystyle~{}~{}\mbox{for}~{}~{}k=0~{}~{}$ (10a)
and $\displaystyle\left\langle{Z}\right\rangle={\rm v}/\sqrt{2}$
$\displaystyle~{}~{}\mbox{for}~{}~{}k\neq 0.$ (10b)

Hereafter, the value of a quantity $Q$ for both alternatives above, – i.e.
either along the flat direction of Eq. (10a) or at the (stable) minimum of Eq.
(10b) – is denoted by the same symbol $\left\langle{Q}\right\rangle$.

Applying Eq. (98) for $K=K_{\rm f}$, and an unknown $W=W_{\rm f}$ for
$Z={Z^{*}}$, we obtain

$V_{\rm f}=e^{Z^{2}}\left(\left(ZW_{\rm f}+W_{\rm
f}^{\prime}\right)^{2}-3W_{\rm f}^{2}\right).$ (11)

Following the strategy in Sec. II, we first find the required form of $W_{\rm
f}$, $W_{0\rm f}$, which assures the establishment of a $Z$-flat direction
with Minkowski vacua. I.e. we require $\left\langle{V_{\rm f}}\right\rangle=0$
for any $Z$. Solving the resulting ordinary differential equation

$Z+\frac{dW_{0{\rm f}}}{dZW_{0\rm f}}=\pm\sqrt{3}$ (12)

w.r.t $W_{0\rm f}$, we obtain two possible forms of $W_{0\rm f}$,

$W_{0\rm f}^{\pm}(Z)=mwf^{\pm
1}~{}~{}\mbox{with}~{}~{}w=e^{-Z^{2}/2}~{}~{}\mbox{and}~{}~{}f=e^{\sqrt{3}Z}.$
(13)

Note that a factor $w$ appears already in the models of Ref. noscale19
associated, though, with a matter field and not with the goldstino superfield
as in our case.

The solutions in Eq. (13) above can be combined as follows – cf. Eq. (5) –

$W_{\Lambda\rm f}=C^{+}W_{0\rm f}^{+}-C^{-}W_{0\rm f}^{-}=mwfC_{f}^{-},$ (14)

where we introduce the symbols

$C_{f}^{\pm}:=C^{+}\pm C^{-}f^{-2}.$ (15)

Employing $K=K_{\rm f}$ and $W=W_{\Lambda\rm f}$ from Eqs. (8) and (14), we
find the corresponding $V$ via Eq. (98)

$\displaystyle V_{\rm f}$ $\displaystyle=\left(m^{2}/(1-12k^{2}Z_{\rm
v}^{2})\right)|f|^{2}\exp\left(-Z_{-}^{2}/2-k^{2}Z_{\rm v}^{4}\right)$ (16)
$\displaystyle\cdot\left(\left|\sqrt{3}C_{f}^{+}-Z_{-}C_{f}^{-}-4k^{2}Z_{\rm
v}^{3}C_{f}^{-}\right|^{2}-3\left|C_{f}^{-}\right|^{2}\right),$

which exhibits the dS/AdS vacua in Eqs. (10a) and (10b). Indeed, we verify
that $\left\langle{V_{\rm f}}\right\rangle=V_{\Lambda}$, given in Eq. (6) and
Eq. (99) for $V=V_{\rm f}$ and ${\alpha}=1$ is readily fulfilled. Indeed,
decomposing $Z_{\alpha}$ (with ${\alpha}=1$ suppressed when we have just one
$Z$) in real and imaginary parts, – $z_{1}:=z$ and ${\bar{z}}_{1}:={\bar{z}}$
– i.e.,

$Z_{\alpha}=(z_{\alpha}+i{\bar{z}}_{\alpha})/\sqrt{2},$ (17)

we find that the eigenvalues of $M_{0}^{2}$ in Eq. (100) are

$\widehat{m}^{2}_{z\rm
f}=144k^{2}m_{3/2f}^{2}\left\langle{{C_{f}^{+}}/{C_{f}^{-}}}\right\rangle^{2}~{}~{}\mbox{and}~{}~{}\widehat{m}_{{\bar{z}}\rm
f}=4m_{3/2f}^{2},$ (18)

where $m_{3/2f}$ is the $\widetilde{G}$ mass along the configurations in Eqs.
(10a) and (10b). This is found by replacing $K$ and $W$ from Eqs. (8) and (14)
in Eq. (103), with result

$\displaystyle m_{3/2f}$
$\displaystyle=m\left\langle{fC_{f}^{-}}\right\rangle$ (19)
$\displaystyle=m\begin{cases}e^{\sqrt{3}Z}\left(C^{+}-C^{-}e^{-2\sqrt{3}Z}\right)&\mbox{for}~{}~{}k=0\\\
e^{\sqrt{3/2}{\rm v}}\left(C^{+}-C^{-}e^{-\sqrt{6}{\rm
v}}\right)&\mbox{for}~{}~{}k\neq 0.\end{cases}$

Note that, for $k=0$ and unfixed $Z$, $m_{3/2f}$ remains undetermined
validating thereby the no-scale character of our models – cf. Ref. old ;
nsreview . As a shown in Eq. (18), the real component $z$ of $Z$ remains
massless due to the flatness of $V_{\rm f}$ along the direction in Eq. (10a).
However, the $k$-dependent term in Eq. (8) not only stabilizes $Z$ but also
provides mass to $z$. On the other hand, this term generates poles and so
discontinuities in $V_{\rm f}$ – see Eq. (6). We are obliged, therefore, to
focus on a local dS/AdS minimum as in Eq. (10b). Inserting Eqs. (18) and (19)
into Eq. (101a) we find

${\sf STr}M_{\rm f}^{2}=\widehat{m}^{2}_{z\rm f}+\widehat{m}^{2}_{{\bar{z}}\rm
f}-4m_{3/2f}^{2}=\widehat{m}^{2}_{z\rm f},$ (20)

which is consistent with Eq. (108) given that ${\cal R}$ in Eq. (109) is found
to be $\left\langle{\mathcal{R}_{\rm f}}\right\rangle=24k^{2}$.

Fig. 1: The (dimensionless) SUGRA potential $10^{2}V_{\rm f}/m^{2}m_{\rm
P}^{2}$ in Eq. (16) as a function of $z$ and ${\bar{z}}$ in Eq. (17) for the
inputs shown in column A of Table 1. The location of the dS vacuum in Eq.
(10b) is also depicted by a thick black point.

Our analytic findings above can be further confirmed by Fig. 1, where the
dimensionless quantity $10^{2}V_{\rm f}/m^{2}m_{\rm P}^{2}$ is plotted as a
function of $z$ and ${\bar{z}}$ in Eq. (17). We employ the values of the
parameters listed in column A of Table 1 – obviously $k_{1}$ there is
identified with $k$ in Eq. (8). We see that the dS vacuum in Eq. (10b) –
indicated by the black thick point – is placed at $(z,{\bar{z}})=(1,0)$ and is
stabilized w.r.t both directions. In the same column of Table 1 displayed are
also the various masses of $\widetilde{G}$ and the scalar ($\widehat{z}$) and
pseudoscalar ($\widehat{\bar{z}}$) components of the sgoldstino $Z$ given in
GeV for convenience. For $N=1$, the spectrum does not comprise any goldstino
$(\widehat{\tilde{z}})$ as explained in Appendix A. It is worth mentioning
that the aforementioned masses may acquire quite natural values (of the order
of $10^{-15}$) for logical values of the relevant parameters despite the fact
that the fulfilment of Eq. (7) via Eq. (6) requires a tiny $C^{-}$. E.g., for
the parameters given in Table 1 we need $C^{-}=1.4\cdot 10^{-90}$.

Let us, finally, note that performing a Kähler transformation

$K\rightarrow K+\Lambda_{K}+\Lambda_{K}^{*}\>\>\>\mbox{and}\>\>\>W\rightarrow
We^{-\Lambda_{K}},$ (21)

with $K=K_{\rm f}$ and $W=W_{\Lambda\rm f}$ in Eqs. (8) and (14) respectively
and $\Lambda_{K}=-Z^{2}/2$, the present model is equivalent with that
described by the following $K$ and $W$

$\widetilde{K}_{\rm f}=-\frac{1}{2}Z_{-}^{2}-k^{2}Z_{\rm
v}^{4}~{}~{}\mbox{and}~{}~{}\widetilde{W}_{\Lambda\rm f}=mfC_{f}^{-}.$ (22)

From the form above we can easily infer that, for $k\rightarrow 0$,
$\widetilde{K}_{\rm f}$ enjoys the enhanced symmetries

$Z\rightarrow Z+c~{}~{}\mbox{and}~{}~{}Z\rightarrow-Z,$ (23)

where $c$ is a real number. These are more structured than the simple $U(1)$
mentioned below Eq. (8) and underline, once more, the naturality of the
possible small $k$ values. In this limit, a similar model arises in the
context of the ${\alpha}$-scale SUGRA introduced in Ref. roest .

Table 1: A Case Study Overview  Case: | A | B | C | D | E | F
---|---|---|---|---|---|---
Input Settings
$K$ | $K_{\rm f}$ | $K_{2\rm f}$ | $K_{\boldsymbol{+}}$ | $K_{\boldsymbol{-}}$ | $K_{\boldsymbol{+}-}$ | $K_{\rm f\boldsymbol{-}}$
$W$ | $W_{\Lambda\rm f}$ | $W_{\Lambda 2\rm f}$ | $W_{\Lambda\boldsymbol{+}}$ | $W_{\Lambda\boldsymbol{-}}$ | $W_{\Lambda\boldsymbol{+}-}$ | $W_{\Lambda\rm f\boldsymbol{-}}$
Input Parameters
${\rm v}_{\alpha}=m_{\rm P}$, $C^{+}=0.01$ and $m=5~{}{\mbox{\rm TeV}}$
$k_{1}$ | $0.3$ | $0.3$ | $0.3$ | $0.3$ | $0.3$ | $0.3$
$k_{2}$ | $-$ | $0.2$ | $-$ | $-$ | $0.2$ | $0.2$
$a_{1}$ | $-$ | $1$ | $-$ | $-$ | $1$ | $1$
$n_{1}$ | $-$ | $-$ | $1$ | $4$ | $1$ | $4$
$n_{2}$ | $-$ | $-$ | $-$ | $-$ | $4$ | $-$
Particle Masses in GeV
$\widetilde{G}$ | $170$ | $276$ | $145$ | $180$ | $263$ | $288$
$\widehat{z}_{1}$ | $612$ | $572$ | $960$ | $530$ | $744$ | $599$
$\widehat{{\bar{z}}}_{1}$ | $340$ | $551$ | $581$ | $180$ | $1005$ | $577$
$\widehat{z}_{2}$ | $-$ | $540$ | $-$ | $-$ | $422$ | $528$
$\widehat{{\bar{z}}}_{2}$ | $-$ | $551$ | $-$ | $-$ | $372$ | $466$
$\widehat{\tilde{z}}_{1}$ | $-$ | $276$ | $-$ | $-$ | $263$ | $288$

###  B Multi-Modular Model

The model above can be extended to incorporate more than one modulus. In this
case, the corresponding $K$ is written as

$K_{N\rm f}=\sum_{\ell=1}^{N_{\rm f}}\left(|Z_{\ell}|^{2}-k_{\ell}^{2}Z_{{\rm
v}\ell}^{4}\right)$ (24)

where for any modulus $Z_{{\alpha}}$ we include a stabilization term

$Z_{\rm v{\alpha}}=Z_{{\alpha}+}-\sqrt{2}{\rm
v}_{\alpha}~{}~{}\mbox{with}~{}~{}Z_{{\alpha}\pm}=Z_{\alpha}\pm
Z^{*}_{\alpha},$ (25)

with ${\alpha}=\ell$ in the domain of the values shown in Eq. (24). As we
verify below, for the same ${\alpha}$ values, we can obtain the stable
configurations

$\displaystyle Z_{{\alpha}-}=0$
$\displaystyle~{}~{}\mbox{for}~{}~{}k_{\alpha}=0$ (26a) and
$\displaystyle\left\langle{Z_{\alpha}}\right\rangle={\rm v}_{\alpha}/\sqrt{2}$
$\displaystyle~{}~{}\mbox{for}~{}~{}k_{\alpha}\neq 0.$ (26b)

Along them the Kähler metric is represented by a $N_{\rm f}\times N_{\rm f}$
diagonal matrix

$\left\langle{K_{{\alpha}{\bar{\beta}}}}\right\rangle={\sf diag}(1,...,1).$
(27)

Setting $Z_{\ell}=Z_{\ell}^{*}$, $K=K_{N\rm f}$ from Eq. (24) and $W=W_{N\rm
f}(Z_{\ell})$ in Eq. (98), $V$ takes the form

$V_{N\rm
f}=e^{\sum_{\ell}Z_{\ell}^{2}}\left(\mbox{$\sum_{\ell}$}\left(Z_{\ell}W_{N\rm
f}+\partial_{\ell}W_{N\rm f}\right)^{2}-3W_{N\rm f}^{2}\right).$ (28)

Setting $V_{N\rm f}=0$ and assuming the following form for the corresponding
$W_{N\rm f}$

$\displaystyle W_{0N\rm f}(Z_{1},...,Z_{N_{\rm f}})$ $\displaystyle=$
$\displaystyle\prod_{\ell}W_{{\rm f}\ell}(Z_{\ell})\Rightarrow$
$\displaystyle\partial_{\ell}W_{0N\rm f}$ $\displaystyle=$
$\displaystyle{dW_{{\rm f}\ell}\over dZ_{\ell}}{W_{0N\rm f}\over W_{{\rm
f}\ell}},$ (29)

we obtain the separated differential equations

$\sum_{\ell}\left(Z_{\ell}+\frac{dW_{{\rm f}\ell}}{dZ_{\ell}W_{{\rm
f}\ell}}\right)^{2}=3.$ (30)

These can be solved w.r.t $W_{{\rm f}\ell}$, if we set

$Z_{\ell}+\frac{dW_{{\rm f}\ell}}{dZ_{\ell}W_{{\rm
f}\ell}}=|a_{\ell}|~{}~{}\mbox{with}~{}~{}\sum_{\ell}a_{\ell}^{2}=3,$ (31)

i.e., the $a_{\ell}$’s satisfy the equation of the hypersphere
$\mathbb{S}^{N_{\rm f}-1}$ with radius $\sqrt{3}$. The resulting solutions
take the form

$W_{{\rm
f}\ell}^{\pm}=w_{\ell}f_{\ell}^{\pm{a_{\ell}}/{\sqrt{3}}}~{}~{}\mbox{with}~{}~{}w_{\ell}=e^{-Z_{\ell}^{2}/2}~{}~{}\mbox{and}~{}~{}f_{\ell}=e^{\sqrt{3}Z_{\ell}}.$
(32)

The total expression for $W_{0N\rm f}$ is found substituting the findings
above into Eq. (29). Namely,

$W_{0N\rm f}^{\pm}=m\prod_{\ell}W_{{\rm f}\ell}^{\pm}=m{\cal W}F^{\pm 1},$
(33)

where we define the functions

${\cal
W}=e^{-\sum_{\ell}Z_{\ell}^{2}/2}~{}~{}\mbox{and}~{}~{}F=e^{\sum_{\ell}a_{\ell}Z_{\ell}}.$
(34)

As in the case with $N_{\rm f}=1$, we combine both solutions above as follows

$\displaystyle W_{\Lambda N\rm f}=C^{+}W_{0N\rm f}^{+}-C^{-}W_{0N\rm
f}^{-}=m{\cal W}FC_{F}^{-},$ (35)

where we introduce the “generalized” $C$ symbols – cf. Eq. (15)

$C_{F}^{\pm}:=C^{+}\pm C^{-}F^{-2}.$ (36)

Substituting Eqs. (24) and (35) into Eq. (98) we find that $V$ takes the form

$\displaystyle V_{N\rm f}$
$\displaystyle=m^{2}|F|^{2}\exp\left(-\sum_{\ell}\left(Z_{\ell-}^{2}/2+k_{\ell}^{2}Z_{{\rm
v}\ell}^{4}\right)\right)$ (37)
$\displaystyle\cdot\left(\sum_{\ell}\frac{\left|a_{\ell}C_{F}^{+}-Z_{\ell-}C_{F}^{-}-4k_{\ell}^{2}Z_{{\rm
v}\ell}^{3}C_{F}^{-}\right|^{2}}{1-12k_{\ell}^{2}Z_{{\rm
v}\ell}^{2}}-3\left|C_{F}^{-}\right|^{2}\right).$

We can confirm that $V_{N\rm f}$ admits the dS/AdS vacua in Eqs. (26a) and
(26b) for ${\alpha}=\ell$, since $\left\langle{V_{N\rm
f}}\right\rangle=V_{\Lambda}$, given in Eq. (6). In addition, Eq. (99) for
$V=V_{N\rm f}$ and ${\alpha}=\ell$ is satisfied, since the $2N_{\rm f}$ masses
squared of the relevant matrix in Eq. (100) are positive. Indeed, analyzing
$Z_{\ell}$ in real and imaginary parts as in Eq. (17), we find

$\displaystyle\widehat{m}_{z\rm f\ell}^{2}$ $\displaystyle=$ $\displaystyle
48k_{\ell}^{2}a_{\ell}^{2}m_{3/2F}^{2}\left\langle{{C_{F}^{+}}/{C_{F}^{-}}}\right\rangle^{2};$
(38a) $\displaystyle\widehat{m}_{\bar{z}\rm f\ell}^{2}$ $\displaystyle=$
$\displaystyle
4m_{3/2F}^{2}\left(1+(3-a_{\ell}^{2})V_{\Lambda}/6{m_{3/2F}^{2}}\right),$
(38b)

where $m_{3/2}$ for the present case is computed inserting Eqs. (24) and (35)
into Eq. (103) with result

$\displaystyle\hskip
56.9055ptm_{3/2F}=m\left\langle{FC_{F}^{-}}\right\rangle=$ (39) $\displaystyle
m\cdot\begin{cases}e^{\sum_{\ell}a_{\ell}Z_{\ell}}\left(C^{+}-C^{-}e^{-2\sum_{\ell}a_{\ell}Z_{\ell}}\right)&\mbox{for}~{}~{}k_{\ell}=0\\\
e^{\sum_{\ell}a_{\ell}\frac{{\rm
v}_{\ell}}{\sqrt{2}}}\left(C^{+}-C^{-}e^{-\sqrt{2}\sum_{\ell}a_{\ell}{\rm
v}_{\ell}}\right)&\mbox{for}~{}~{}k_{\ell}\neq 0.\end{cases}$

The expressions above conserve the basic features of the no-scale models as
explained below Eq. (19). We consider the stabilized version of these models
(with $k_{\alpha}\neq 0$) as more complete since it offers the determination
of $m_{3/2}$ and avoids the presence of a massless mode which may be
problematic.

We should note that the relevant $2N_{\rm f}\times 2N_{\rm f}$ matrix
$M_{0}^{2}$ of Eq. (100) turns out to be diagonal up to some tiny mixings
appearing in the ${\bar{z}}_{\ell}-{\bar{z}}_{\bar{\ell}}$ positions. These
contributions though can be safely neglected since these are proportional to
$V_{\Lambda}$. We also obtain $N_{\rm f}-1$ Weyl fermions with masses
$\widehat{m}_{\tilde{z}\tilde{\ell}}=m_{3/2}$ where $\tilde{\ell}=1,...,N_{\rm
f}-1$. Note that Eqs. (39), (38a) and (38b) reduce to the ones obtained for
$N_{\rm f}=1$, i.e. Eqs. (19) and (18), if we replace $a_{\ell}=\sqrt{3}$.
Inserting the mass spectrum above into Eq. (101a), we find

${\sf STr}M_{N\rm f}^{2}=2(N_{\rm
f}-1)\left(m_{3/2F}^{2}+V_{\Lambda}\right)+\mbox{$\sum_{\ell}$}\widehat{m}^{2}_{z\rm
f\ell}.$ (40)

This result is consistent with Eq. (101b) given that its last term turns out
to be equal to the last term of Eq. (40).

To highlight further the conclusions above we depict in Fig. 2 the
dimensionless $V_{N\rm f}$ for $N=2$, i.e. $V_{2\rm f}$, as a function of
$z_{1}$ and $z_{2}$ for ${\bar{z}}_{1}={\bar{z}}_{2}=0$ and the other
parameters displayed in column B of Table 1. We observe that the dS vacuum in
Eq. (26b) – indicated by a black thick point in the plot – is well stabilized
against both directions. In the same column of Table 1 we also arrange some
suggestive values of the particle masses for $N=2$. Note that, due to the
smallness of $V_{\Lambda}$, the $\widehat{m}_{{\bar{z}}\rm f\ell}$ values are
practically equal between each other.

Fig. 2: The (dimensionless) SUGRA potential $V_{2\rm f}/m^{2}m_{\rm P}^{2}$ in
Eq. (37) as a function of $z_{1}$ and $z_{2}$ in Eq. (17) for
${\bar{z}}_{1}={\bar{z}}_{2}=0$ and the inputs shown in column B of Table 1.
The location of the dS vacuum in Eq. (26b) is also depicted by a thick black
point.

##  IV Curved Moduli Geometry

We proceed now to the models with curved internal geometry and describe below
their version for one – see Sec. A – or more – see Sec. B – moduli.

###  A Uni-Modular Model

The curved moduli geometry is described mainly by the Kähler potentials

$K_{\boldsymbol{\pm}}=\pm
n\ln\Omega_{\boldsymbol{\pm}}~{}~{}\mbox{with}~{}~{}\Omega_{\boldsymbol{\pm}}=1\pm\frac{|Z|^{2}-k^{2}Z_{\rm
v}^{4}}{n}$ (41)

and $Z_{\rm v}$ given in Eq. (9). Also $n>0$ and $k$ are real, free
parameters. The positivity of the argument of logarithm in Eq. (41) implies

$1\pm|Z|^{2}/n\gtrsim
0~{}~{}\Rightarrow\begin{cases}|Z|^{2}\gtrsim-n&\mbox{for}~{}~{}K=K_{\boldsymbol{+}},\\\
|Z|\lesssim\sqrt{n}&\mbox{for}~{}~{}K=K_{\boldsymbol{-}}.\end{cases}$ (42)

The restriction for $K=K_{\boldsymbol{+}}$ is trivially satisfied, whereas
this for $K=K_{\boldsymbol{-}}$ defines the allowed domain of $Z$ values which
lie in a disc with radius $\sqrt{n}$ and thus, the name disk coordinates. If
we set $k=0$ in Eq. (41), $K_{\boldsymbol{-}}$ parameterizes noscale ; su11 ;
nsreview  the coset space $SU(1,1)/U(1)$ whereas $K_{\boldsymbol{+}}$ is
associated su11  with $SU(2)/U(1)$. Thanks to these symmetries, low $k$ values
are totally natural as we explained below Eq. (23). The Kähler metric and (the
constant) $\mathcal{R}$ in Eq. (109) are respectively

$K_{ZZ^{*}}=\Omega_{\boldsymbol{\pm}}^{-2}~{}~{}\mbox{and}~{}~{}\mathcal{R}_{\rm\pm}=\pm
2/n~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{\pm}}.$ (43)

The last quantity reveals that the Kähler manifold is compact (spherical) or
non-compact (hyperbolic) if $K=K_{\boldsymbol{+}}$ or $K=K_{\boldsymbol{-}}$
respectively. For this reason, the bold subscripts ${\boldsymbol{+}}$ or
${\boldsymbol{-}}$ associated with various quantities below are referred to
$K=K_{\boldsymbol{+}}$ or $K=K_{\boldsymbol{-}}$ respectively.

Repeating the procedure described in Sec. II, we find the form of $V$ in Eq.
(98), $V_{\boldsymbol{\pm}}$, as a function of $K=K_{\boldsymbol{\pm}}$ in Eq.
(41) and $W=W_{\boldsymbol{\pm}}$ for $Z={Z^{*}}$. This is

$V_{\boldsymbol{\pm}}=v_{\boldsymbol{\pm}}^{\pm
n}W_{\boldsymbol{\pm}}^{2}\left(\left(Z+v_{\boldsymbol{\pm}}\frac{W_{\boldsymbol{\pm}}^{\prime}}{W_{\boldsymbol{\pm}}}\right)^{2}-3\right)~{}\mbox{with}~{}~{}v_{\boldsymbol{\pm}}=1\pm\frac{Z^{2}}{n}.$
(44)

Setting $V_{\boldsymbol{\pm}}=0$ we see that the corresponding
$W_{\boldsymbol{\pm}}=W_{0\boldsymbol{\pm}}$ obeys the differential equation

$\frac{dW_{0\boldsymbol{\pm}}}{W_{0\boldsymbol{\pm}}}=\frac{\pm\sqrt{3}-Z}{v_{\boldsymbol{\pm}}}dZ.$
(45)

This can be resolved yielding two possible forms of $W_{0\boldsymbol{\pm}}$,

$W_{0\boldsymbol{\pm}}^{\pm}=m{v}^{\mp
n/2}_{\boldsymbol{\pm}}u_{\boldsymbol{\pm}}^{\pm
1}~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{\pm}},$ (46)

which assure the establishment of Minkowski minima – cf. Eq. (3). The
corresponding functions $u_{\boldsymbol{\pm}}$ can be specified as follows

$u_{\boldsymbol{+}}=e^{\sqrt{3n}{\rm
atn}(Z/n)}~{}~{}\mbox{and}~{}~{}u_{\boldsymbol{-}}=e^{\sqrt{3n}{\rm
atnh}(Z/n)},$ (47)

where ${\rm atn}$ and ${\rm atnh}$ stand for the functions $\arctan$ and ${\rm
arctanh}$ respectively. The superscript $\pm$ in Eq. (46) correspond to the
exponents of $u_{\boldsymbol{\pm}}$ and should not be confused with the bold
subscripts ${\boldsymbol{\pm}}$ with reference to $K_{\boldsymbol{\pm}}$.

Combining both Minkowski solutions, $W_{0\boldsymbol{\pm}}^{\pm}$ in Eq. (46)
and introducing the shorthand notation – cf. Eq. (15) –

$C_{u\boldsymbol{\pm}}^{\pm}:=C^{+}\pm
C^{-}u_{\boldsymbol{\pm}}^{-2}~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{\pm}},$
(48)

we can obtain the superpotential

$\displaystyle W_{\Lambda\boldsymbol{\pm}}$
$\displaystyle=C^{+}W_{0\boldsymbol{\pm}}^{+}-C^{-}W_{0\boldsymbol{\pm}}^{-}$
(49) $\displaystyle=mv_{\boldsymbol{\pm}}^{\mp
n/2}u_{\boldsymbol{\pm}}C_{u\boldsymbol{\pm}}^{-}~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{\pm}}$

which allows for dS/AdS vacua. To verify it, we insert
$K=K_{\boldsymbol{\pm}}$ and $W=W_{\rm\Lambda\boldsymbol{\pm}}$ from Eqs. (41)
and (49) in Eq. (98) with result

$\displaystyle V_{\boldsymbol{\pm}}$
$\displaystyle=m^{2}\Omega_{\boldsymbol{\pm}}^{\pm
n}|v_{\boldsymbol{\pm}}|^{\mp
n}|u_{\boldsymbol{\pm}}|^{2}\Big{(}-3|C_{u\boldsymbol{\pm}}^{-}|^{2}+n\Omega_{\boldsymbol{\pm}}^{2}$
(50)
$\displaystyle\cdot\left|(\sqrt{3}C_{u\boldsymbol{\pm}}^{+}-ZC_{u\boldsymbol{\pm}}^{-})v_{\boldsymbol{\pm}}^{-1}\pm(Z^{*}-4k^{2}Z_{\rm
v}^{3})C_{u\boldsymbol{\pm}}^{-}\Omega_{\boldsymbol{\pm}}^{-1}\right|^{2}$
$\displaystyle\cdot\left(\mp|4k^{2}Z_{\rm
v}^{3}-Z|^{2}+n\Omega_{\boldsymbol{\pm}}(1-12k^{2}Z_{\rm
v}^{2})\right)^{-1}\Big{)}.$

Given that for $Z_{-}=0$ we get
$\Omega_{\boldsymbol{\pm}}=v_{\boldsymbol{\pm}}$, we may infer that
$\left\langle{V_{\boldsymbol{\pm}}}\right\rangle=V_{\Lambda}$ shown in Eq. (6)
for the directions in Eqs. (10a) and (10b). Eq. (99) for
$V=V_{\boldsymbol{\pm}}$ and $Z_{1}:=Z$ is also valid without restrictions for
$K=K_{\boldsymbol{+}}$ but only for $n>3$ for $K=K_{\boldsymbol{-}}$. In fact,
employing the decomposition of $Z$ in Eq. (17) for ${\alpha}=1$, we can obtain
the scalar spectrum of our models which includes the sgoldstino components
with masses squared

$\displaystyle\widehat{m}_{z\boldsymbol{\pm}}^{2}$ $\displaystyle=$
$\displaystyle
144k^{2}m_{3/2\boldsymbol{\pm}}^{2}\left\langle{v_{\boldsymbol{\pm}}^{3/2}\left|{C_{u\boldsymbol{\pm}}^{+}}/{C_{u\boldsymbol{\pm}}^{-}}\right|}\right\rangle^{2};$
(51a) $\displaystyle\widehat{m}_{{\bar{z}}\boldsymbol{\pm}}^{2}$
$\displaystyle=$ $\displaystyle
4m_{3/2\boldsymbol{\pm}}^{2}\left(1\pm(3/n)\left\langle{C_{u\boldsymbol{\pm}}^{+}/C_{u\boldsymbol{\pm}}^{-}}\right\rangle^{2}\right)$
(51b)

for $K=K_{\boldsymbol{\pm}}$ respectively. The corresponding $m_{3/2}$
according to Eq. (103) – with $K$ and $W$ given in Eqs. (41) and (49) – is

$m_{3/2\boldsymbol{\pm}}=m\left\langle{u_{\boldsymbol{\pm}}C_{u\boldsymbol{\pm}}^{-}}\right\rangle~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{\pm}},$
(52)

which may be explicitly written if we use Eqs. (47) and (48) – cf. Eq. (19).
The stability of configurations in Eqs. (10a) and (10b) is protected for
$\widehat{m}_{{\bar{z}}\boldsymbol{-}}^{2}>0$ and
$\widehat{m}_{{\bar{z}}\boldsymbol{-}}^{2}>0$ provided that

$Z<\sqrt{n}~{}~{}\mbox{and}~{}~{}n>3~{}~{}\mbox{for}~{}~{}K=K_{\boldsymbol{-}}.$
(53)

Since we expect that $Z\leq 1$, the latter restriction is capable to
circumvent both requirements – see column D in Table 1. Inserting the mass
spectrum above into the definition of Eq. (101a), we can find

${\sf
STr}M_{\boldsymbol{\pm}}^{2}=12m_{3/2\boldsymbol{\pm}}^{2}\left\langle{C_{u\boldsymbol{\pm}}^{+}/C_{u\boldsymbol{\pm}}^{-}}\right\rangle^{2}\left(12nk^{2}\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{3}\pm
1\right).$ (54)

It can be easily verified that the result above is consistent with the
expression of Eq. (108) given that $\mathcal{R}$ in Eq. (109) is

$\left\langle{\mathcal{R}_{\boldsymbol{\pm}}}\right\rangle=2\left(12k^{2}n\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{3}\pm
1\right).$ (55)

Fig. 3: The (dimensionless) SUGRA potential $V/m^{2}m_{\rm P}^{2}$ as a
function of $z$ for ${\bar{z}}=0$ and the settings in columns A (solid line),
C (dot-dashed line) and D (dashed line) of Table 1. The value
$\left\langle{z}\right\rangle=m_{\rm P}$ is also indicated.

Our analytic results are exemplified in Fig. 3, where we depict
$V_{\boldsymbol{+}}$ (dot-dashed line) $V_{\boldsymbol{-}}$ (dashed line)
together with $V_{\rm f}$ (solid line) versus $z$ for ${\bar{z}}=0$,
$k=k_{1}=0.3$ and the other parameters shown in columns A, C and D of Table 1.
Note that the selected $n=n_{1}=4$ for $K=K_{\boldsymbol{-}}$ protects the
stability of the vacuum in Eq. (10b) as dictated above. In columns C and D of
Table 1 we display some explicit values of the particle masses encountered for
$K=K_{\boldsymbol{-}}$ and $K_{\boldsymbol{+}}$ respectively. As a consequence
of the employed $n$ value in column D we accidentally obtain
$m_{3/2\boldsymbol{-}}=\widehat{m}_{{\bar{z}}1}=\widehat{m}_{{\bar{z}}-}$;
${k_{1}}$ and $n_{1}$ obviously coincide with $k$ and $n$ in Eq. (41).

###  B Multi-Modular Model

The generalization of the model above to incorporate more than one modulus can
be performed following the steps of Sec. B. This generalization, however, is
accompanied by a possible mixing of the two types of the curved geometry
analyzed in Sec. A. More specifically, the considered here $K$, includes two
sectors with $N_{\boldsymbol{+}}$ compact components and $N_{\boldsymbol{-}}$
non-compact ones. It may be written as

$K_{N_{\boldsymbol{+}}N_{\boldsymbol{-}}}=\sum_{i=1}^{N_{\boldsymbol{+}}}n_{i}\ln\Omega_{i}+\sum_{j=N_{\boldsymbol{+}}+1}^{N_{\boldsymbol{\pm}}}n_{j}\ln\Omega_{j},$
(56)

where $N_{\boldsymbol{\pm}}=N_{\boldsymbol{+}}+N_{\boldsymbol{-}}$ and the
arguments of the logarithms are identified as

$\Omega_{\alpha}=\begin{cases}\Omega_{{\alpha}\boldsymbol{+}}&~{}~{}\mbox{for}~{}~{}{\alpha}=i,\\\
\Omega_{{\alpha}\boldsymbol{-}}&~{}~{}\mbox{for}~{}~{}{\alpha}=j.\end{cases}$
(57a) The symbols $\Omega_{{\alpha}\boldsymbol{\pm}}$ can be collectively
defined as
$\Omega_{{\alpha}\boldsymbol{\pm}}=1\pm|Z_{\alpha}|^{2}/n_{\alpha}\mp
k_{\alpha}^{2}Z_{\rm v{\alpha}}^{4}$ (57b)

with $Z_{{\rm v}{\alpha}}$ is given from Eq. (25) and ${\alpha}=i,j$. When
explicitly indicated, summation and multiplication over $i$ and $j$ is applied
for the range of their values specified in Eq. (56). Given that $i$
corresponds to compact geometry (${\boldsymbol{+}}$) and $j$ to non-compact
(${\boldsymbol{-}}$) we remove the relevant indices ${\boldsymbol{\pm}}$ from
the various quantities to simplify the notation. Under these assumptions, the
positivity of the arguments of $\ln$ implies restrictions only to $\Omega_{j}$
– cf. Eq. (42):

$\Omega_{j}>0~{}~{}\Rightarrow~{}~{}|Z_{j}|<\sqrt{n_{j}}\,.$ (58)

Along the configurations in Eqs. (26a) and (26b) for ${\alpha}=i,j$, the
Kähler metric is represented by a $N_{\boldsymbol{\pm}}\times
N_{\boldsymbol{\pm}}$ diagonal matrix

$K_{{\alpha}{\bar{\beta}}}={\sf
diag}(v_{1}^{-2},...,v_{N_{+}}^{-2},v_{N_{+}+1}^{-2},...,v_{N_{\boldsymbol{\pm}}}^{-2}),$
(59)

where we introduce the generalizations of the symbols $v_{\boldsymbol{\pm}}$,
defined in Eq. (44), as follows

$v_{i}=(1+Z_{i}^{2}/n_{i})~{}~{}\mbox{and}~{}~{}v_{j}=(1-Z_{j}^{2}/n_{j}).$
(60)

Also $\mathcal{R}$ in Eq. (109) includes contributions for both geometric
sectors, i.e,

$\mathcal{R}_{N_{\boldsymbol{+}}N_{\boldsymbol{-}}}=2\mbox{$\sum_{i}$}n_{i}^{-1}-2\mbox{$\sum_{j}$}n_{j}^{-1}.$
(61)

Inserting $K=K_{N_{\boldsymbol{+}}N_{\boldsymbol{-}}}$ from Eq. (56) and
$W=W_{N\boldsymbol{\pm}}(Z_{i},Z_{j})$ with $Z_{i}=Z^{*}_{i}$ and
$Z_{j}=Z^{*}_{j}$ in Eq. (98), we obtain

$\displaystyle V_{N\boldsymbol{\pm}}$ $\displaystyle=$
$\displaystyle\mathcal{V}^{-2}\Big{(}\mbox{$\sum_{i}$}\left(Z_{i}W_{N\boldsymbol{\pm}}+\partial_{i}W_{N\boldsymbol{\pm}}v_{i}\right)^{2}$
(62) $\displaystyle+$
$\displaystyle\mbox{$\sum_{j}$}\left(Z_{j}W_{N\boldsymbol{\pm}}+\partial_{j}W_{N\boldsymbol{\pm}}v_{j}\right)^{2}-3W_{N\boldsymbol{\pm}}^{2}\Big{)},$

where the prefactor $\mathcal{V}$ is defined as follows

$\mathcal{V}=\mbox{$\prod_{i,j}$}v_{i}^{-n_{i}/2}v_{j}^{n_{j}/2}.$ (63)

Setting $V_{N\boldsymbol{\pm}}=0$ and assuming the ansatz for the
corresponding $W_{N\boldsymbol{\pm}}$

$\displaystyle
W_{0N\boldsymbol{\pm}}(Z_{1},...,Z_{N_{\boldsymbol{\pm}}})=\prod_{i,j}W_{i}(Z_{i})W_{j}(Z_{j})~{}~{}\Rightarrow~{}~{}$
$\displaystyle\frac{\partial_{i}W_{0N\boldsymbol{\pm}}}{W_{0N\boldsymbol{\pm}}}={dW_{i}\over
dZ_{i}W_{i}}~{}~{}\mbox{and}~{}~{}\frac{\partial_{j}W_{0N\boldsymbol{\pm}}}{W_{0N\boldsymbol{\pm}}}={dW_{j}\over
dZ_{j}W_{j}},$ (64)

we obtain the separated differential equations

$\sum_{i}\left(Z_{i}+\frac{dW_{i}}{dZ_{i}W_{i}}v_{i}\right)^{2}+\sum_{j}\left(Z_{j}+\frac{dW_{j}}{dZ_{j}W_{j}}v_{j}\right)^{2}=3.$
(65)

We can solve the equations above if we set

$Z_{i}+\frac{dW_{i}}{dZ_{i}W_{i}}v_{i}=|a_{i}|~{}~{}\mbox{and}~{}~{}Z_{j}+\frac{dW_{j}}{dZ_{j}W_{j}}v_{j}=|a_{j}|$
(66)

imposing the constraint

$\mbox{$\sum_{i}$}a_{i}^{2}+\mbox{$\sum_{j}$}a_{j}^{2}=3,$ (67)

i.e., the $a_{i}$ and $a_{j}$ can be regarded as coordinates of the
hypersphere $\mathbb{S}^{N_{\boldsymbol{\pm}}-1}$ with radius $\sqrt{3}$.
Solution of the differential equations above w.r.t $W_{i}$ and $W_{j}$ yields

$W_{i}^{\pm}=v_{i}^{-n_{i}/2}u_{i}^{\pm
a_{i}/\sqrt{3}}~{}~{}\mbox{and}~{}~{}W_{j}^{\pm}=v_{j}^{n_{i}/2}u_{j}^{\pm
a_{j}/\sqrt{3}}$ (68)

with the generalizations of $u_{\boldsymbol{+}}$ and $u_{\boldsymbol{-}}$ in
Eq. (47) defined as

$u_{i}=e^{\sqrt{3n_{i}}{\rm atn}(Z_{i}/\sqrt{n_{i}})}\\\
~{}~{}\mbox{and}~{}~{}u_{j}=e^{\sqrt{3n_{j}}{\rm atnh}(Z_{j}/\sqrt{n_{j}})}.$
(69)

Upon substitution of Eq. (68) into Eq. (64) we obtain

$W_{0N\boldsymbol{\pm}}^{\pm}=m\prod_{i,j}W_{i}^{\pm}W_{j}^{\pm}=m\mathcal{V}\mathcal{U}^{\pm
1},$ (70)

where we define the function

$\mathcal{U}=\mbox{$\prod_{i,j}$}u_{i}^{a_{i}/\sqrt{3}}u_{j}^{a_{j}/\sqrt{3}}.$
(71)

Introducing the generalized $C$ symbols – cf. Eq. (15) –

$C_{\mathcal{U}}^{\pm}:=C^{+}\pm C^{-}\mathcal{U}^{-2},$ (72)

we combine both solutions in Eq. (70) as follows

$\displaystyle W_{\Lambda
N\boldsymbol{\pm}}=C^{+}W_{0N\boldsymbol{\pm}}^{+}-C^{-}W_{0N\boldsymbol{\pm}}^{-}=m\mathcal{V}\mathcal{U}C_{\mathcal{U}}^{-}.$
(73)

Plugging $K=K_{N_{\boldsymbol{+}}N_{\boldsymbol{-}}}$ and $W=W_{\Lambda
N\boldsymbol{\pm}}$ from Eqs. (56) and (73) in Eq. (98) we find

$\displaystyle V_{N\boldsymbol{\pm}}$
$\displaystyle=m^{2}\left(\mbox{$\prod_{ij}$}\Omega_{i}^{n_{i}}\Omega_{j}^{-n_{j}}\right)|\mathcal{V}\mathcal{U}|^{2}\Bigg{(}-3|C_{\mathcal{U}}^{-}|^{2}$
(74)
$\displaystyle+\sum_{i}\Big{(}n_{i}\Omega_{i}^{2}\cdot\left|(a_{i}C_{\mathcal{U}}^{+}-Z_{i}C_{\mathcal{U}}^{-})v_{i}^{-1}\right.$
$\displaystyle+\left.(Z^{*}_{i}-4k_{i}^{2}Z_{{\rm
v}i}^{3})C_{\mathcal{U}}^{-}\Omega_{i}^{-1}\right|^{2}$
$\displaystyle\cdot\left(-\left|4k_{i}^{2}Z_{{\rm
v}i}^{3}-Z_{i}\right|^{2}+n_{i}\Omega_{i}\left(1-12k_{i}^{2}Z_{{\rm
v}i}^{2}\right)\right)^{-1}\Big{)}$
$\displaystyle+\sum_{j}\Big{(}n_{j}\Omega_{j}^{2}\cdot\left|(a_{j}C_{\mathcal{U}}^{+}-Z_{j}C_{\mathcal{U}}^{-})v_{j}^{-1}\right.$
$\displaystyle-\left.(Z^{*}_{j}-4k_{j}^{2}Z_{{\rm
v}j}^{3})C_{\mathcal{U}}^{-}\Omega_{j}^{-1}\right|^{2}$
$\displaystyle\cdot\left(\left|4k_{j}^{2}Z_{{\rm
v}j}^{3}-Z_{j}\right|^{2}+n_{j}\Omega_{j}\left(1-12k_{j}^{2}Z_{{\rm
v}j}^{2}\right)\right)^{-1}\Big{)}\Bigg{)}.$

Note that there are slight differences between the terms with subscripts $i$
and $j$ due to our convention in Eq. (57a) – cf. Eq. (50). The settings in
Eqs. (26a) and (26b) consist honest dS/AdS vacua since
$\left\langle{V_{N\boldsymbol{\pm}}}\right\rangle=V_{\Lambda}$ given in Eq.
(6). However, the conditions in Eq. (99) for $V=V_{N\boldsymbol{\pm}}$ and
${\alpha}=i,j$ are met only after imposing upper bound on ${\rm v}_{j}$ and
$a_{j}$. To determine this, we extract the masses squared of the
$2N_{\boldsymbol{\pm}}$ scalar components of $Z_{i}$ and $Z_{j}$ in Eq. (17)
which are

$\displaystyle\widehat{m}_{zi\boldsymbol{+}}^{2}$ $\displaystyle=$
$\displaystyle 48k_{i}^{2}a_{i}^{2}m_{3/2{\cal
U}}^{2}\left\langle{v_{i}^{3/2}{C_{\mathcal{U}}^{+}}/{C_{\mathcal{U}}^{-}}}\right\rangle^{2};$
(75a) $\displaystyle\widehat{m}_{zj\boldsymbol{-}}^{2}$ $\displaystyle=$
$\displaystyle 48k_{j}^{2}a_{j}^{2}m_{3/2{\cal
U}}^{2}\left\langle{v_{j}^{3/2}{C_{\mathcal{U}}^{+}}/{C_{\mathcal{U}}^{-}}}\right\rangle^{2};$
(75b) $\displaystyle\widehat{m}_{{\bar{z}}i\boldsymbol{+}}^{2}$
$\displaystyle\simeq$ $\displaystyle 4m_{3/2{\cal
U}}^{2}\left(1+a_{i}^{2}/n_{i}\right);$ (75c)
$\displaystyle\widehat{m}_{{\bar{z}}j\boldsymbol{-}}^{2}$
$\displaystyle\simeq$ $\displaystyle 4m_{3/2{\cal
U}}^{2}\left(1-a_{j}^{2}/n_{j}\right),$ (75d)

where we restore the $\boldsymbol{\pm}$ symbols for clarity and we neglect for
simplicity terms of order $(C^{-})^{2}$ in the two last expressions. We also
compute $m_{3/2}$ upon substitution of Eqs. (56) and (73) into Eq. (103) with
result

$m_{3/2{\cal U}}=m\left\langle{\mathcal{U}C_{\mathcal{U}}^{-}}\right\rangle.$
(76)

As in the case of Sec. B, the relevant matrix $M_{0}^{2}$ in Eq. (100) turns
out to be essentially diagonal since the non-zero elements appearing in the
${\bar{z}}_{\alpha}-{\bar{z}}_{\beta}$ positions with ${\alpha},\beta=i,j$ are
proportional to $V_{\Lambda}$ and can be safely ignored compared to the
diagonal terms. From Eqs. (75b) and (75d), we notice that positivity of
$\widehat{m}_{zj-}^{2}$ and $\widehat{m}_{{\bar{z}}j-}^{2}$ dictates

$Z_{j}<\sqrt{n_{j}}~{}~{}\mbox{and}~{}~{}|a_{j}|<\sqrt{n_{j}}.$ (77)

These restrictions together with Eq. (67) delineate the allowed ranges of
parameters in the hyperbolic sector. We also obtain $N_{\boldsymbol{\pm}}-1$
Weyl fermions with masses $\widehat{m}_{\tilde{z}\tilde{\alpha}}=m_{3/2}$ with
$\tilde{\alpha}=1,...,N_{\boldsymbol{\pm}}-1$. Inserting the mass spectrum
above into Eq. (101a) we find

$\displaystyle{\sf STr}M_{N\boldsymbol{\pm}}^{2}$ $\displaystyle\simeq
6m_{3/2{\cal
U}}^{2}\Bigg{(}(N_{\boldsymbol{\pm}}-1)+\frac{2}{3}\left(\mbox{$\sum_{i}$}\frac{a_{i}^{2}}{n_{i}}+\mbox{$\sum_{j}$}\frac{a_{j}^{2}}{n_{j}}\right)+$
(78) $\displaystyle
8\left\langle{\frac{C_{\mathcal{U}}^{+}}{C_{\mathcal{U}}^{-}}}\right\rangle^{2}\left(\mbox{$\sum_{i}$}a_{i}^{2}k_{i}^{2}\left\langle{v_{i}}\right\rangle^{3}+\mbox{$\sum_{j}$}a_{j}^{2}k_{j}^{2}\left\langle{v_{j}}\right\rangle^{3}\right)\Bigg{)}.$

It can be checked that this result is consistent with Eq. (101b).

Fig. 4: The (dimensionless) SUGRA potential $V_{\boldsymbol{+}-}/m^{2}m_{\rm
P}^{2}$ in Eq. (74) as a function of $z_{1}$ and $z_{2}$ in Eq. (17) for
${\bar{z}}_{1}={\bar{z}}_{2}=0$ and the inputs shown in column E of Table 1.
The location of the dS vacuum in Eq. (26b) is also depicted by a thick black
point.

For $N=2$, $N_{\boldsymbol{-}}=N_{\boldsymbol{+}}=1$ and the parameters shown
in column E of Table 1, we present in Fig. 4 the relevant
$V_{N\boldsymbol{\pm}}$, $V_{2\boldsymbol{\pm}}=V_{\boldsymbol{+}-}$ –
conveniently normalized – versus $z_{1}$ and $z_{2}$ in Eq. (17) fixing
${\bar{z}}_{1}={\bar{z}}_{2}=0$. It is clearly shown that the vacuum of Eq.
(26b), depicted by a bold point, is indeed stable. In column E of Table 1 we
arrange also some representative masses (in GeV) of the particle spectrum for
$N_{\boldsymbol{\pm}}=2$. From the parameters listed there we infer that
$a_{2}=\sqrt{2}<\sqrt{n_{2}}=2$ and so Eqs. (67) and (77) are met.

##  V Generalization

It is certainly impressive that the models described in Sec. B and B can be
combined in a simple and (therefore) elegant way. We here just specify the
utilized $K$ and $W$ of a such model and restrict ourselves to the
verification of the results. In particular, we consider the following $K$

$K_{N\rm f\boldsymbol{\pm}}=K_{N\rm
f}+K_{N_{\boldsymbol{+}}N_{\boldsymbol{-}}},$ (79)

which incorporates the individual contributions from Eqs. (24) and (56). It is
intuitively expected that the required $W$ for achieving dS/AdS vacua has the
form – cf. Eqs. (35) and (73)

$W_{\Lambda N\rm f\boldsymbol{\pm}}=C^{+}W_{0N\rm
f\boldsymbol{\pm}}^{+}+C^{-}W_{0N\rm f\boldsymbol{\pm}}^{-},$ (80)

where the definitions of $W_{0N\rm f\boldsymbol{\pm}}^{\pm}$ follow those in
Eqs. (33) and (70) respectively. Namely, we set

$W_{0N\rm f\boldsymbol{\pm}}^{\pm}=m\mathcal{W}\mathcal{V}(F\mathcal{U})^{\pm
1},$ (81)

where the parameters $a_{\ell},a_{i}$ and $a_{j}$, which enter the expressions
of the functions $F$ and $\mathcal{U}$ in Eqs. (34) and (71), satisfy the
constraint – cf. Eqs. (31) and (67)

$\mbox{$\sum_{\ell}$}a_{\ell}^{2}+\mbox{$\sum_{i}$}a_{i}^{2}+\mbox{$\sum_{j}$}a_{j}^{2}=3.$
(82)

I.e., they lie at the hypersphere $\mathbb{S}^{N_{\rm t}-1}$ with radius
$\sqrt{3}$ and $N_{\rm t}=N_{\rm f}+N_{\boldsymbol{\pm}}$. If we introduce, in
addition, the $C$ symbols – cf. Eqs. (36) and (72) –

$C_{F\mathcal{U}}^{\pm}:=C^{+}\pm C^{-}(F\mathcal{U})^{-2},$ (83)

$W_{\Lambda N\rm f\boldsymbol{\pm}}$ in Eq. (80) is simplified as

$W_{\Lambda N\rm
f\boldsymbol{\pm}}=m\mathcal{W}\mathcal{V}F\mathcal{U}C_{F\mathcal{U}}^{-}.$
(84)

Plugging $K=K_{N\rm f\boldsymbol{\pm}}$ and $W=W_{\Lambda N\rm
f\boldsymbol{\pm}}$ from Eqs. (79) and (84) into Eq. (98) we obtain

$\displaystyle V_{N\rm f\boldsymbol{\pm}}$ $\displaystyle=$ $\displaystyle
m^{2}e^{K_{N\rm
f}}\left(\mbox{$\prod_{ij}$}\Omega_{i}^{n_{i}}\Omega_{j}^{-n_{j}}\right)|F\mathcal{V}\mathcal{U}|^{2}\Bigg{(}-3|C_{F\mathcal{U}}^{-}|^{2}$
(85) $\displaystyle+$
$\displaystyle\sum_{i}\Big{(}n_{i}\Omega_{i}^{2}\cdot\left|(a_{i}C_{F\mathcal{U}}^{+}-Z_{i}C_{F\mathcal{U}}^{-})v_{i}^{-1}\right.$
$\displaystyle+$ $\displaystyle\left.(Z^{*}_{i}-4k_{i}^{2}Z_{{\rm
v}i}^{3})C_{F\mathcal{U}}^{-}\Omega_{i}^{-1}\right|^{2}$ $\displaystyle\cdot$
$\displaystyle\left(-\left|4k_{i}^{2}Z_{{\rm
v}i}^{3}-Z_{i}\right|^{2}+n_{i}\Omega_{i}\left(1-12k_{i}^{2}Z_{{\rm
v}i}^{2}\right)\right)^{-1}\Big{)}$ $\displaystyle+$
$\displaystyle\sum_{j}\Big{(}n_{j}\Omega_{j}^{2}\cdot\left|(a_{j}C_{F\mathcal{U}}^{+}-Z_{j}C_{F\mathcal{U}}^{-})v_{j}^{-1}\right.$
$\displaystyle-$ $\displaystyle\left.(Z^{*}_{j}-4k_{j}^{2}Z_{{\rm
v}j}^{3})C_{F\mathcal{U}}^{-}\Omega_{j}^{-1}\right|^{2}$ $\displaystyle\cdot$
$\displaystyle\left(\left|4k_{j}^{2}Z_{{\rm
v}j}^{3}-Z_{j}\right|^{2}+n_{j}\Omega_{j}\left(1-12k_{j}^{2}Z_{{\rm
v}j}^{2}\right)\right)^{-1}\Big{)}$ $\displaystyle+$
$\displaystyle\sum_{\ell}\left|a_{\ell}C_{F\mathcal{U}}^{+}-Z_{\ell-}C_{F\mathcal{U}}^{-}-4k_{\ell}^{2}Z_{{\rm
v}\ell}^{3}C_{F\mathcal{U}}^{-}\right|^{2}$ $\displaystyle\cdot$
$\displaystyle\left(1-12k_{\ell}^{2}Z_{{\rm v}\ell}^{2}\right)^{-1}\Bigg{)}.$

Once again, we infer that Eqs. (26a) and (26b) consist dS/AdS vacua since
$\left\langle{V_{N\rm f\boldsymbol{\pm}}}\right\rangle=V_{\Lambda}$ – see Eq.
(6) – and Eq. (99) with $V=V_{N\rm f\boldsymbol{\pm}}$ and ${\alpha}=\ell,i,j$
is fulfilled if we take into account the restrictions in Eqs. (77) and (58).

The $\widetilde{G}$ mass is derived from Eq. (103), after substituting $K$ and
$W$ from Eqs. (79) and (84) respectively. The result is

$m_{3/2{\cal
FU}}=m\left\langle{F\mathcal{U}C_{F\mathcal{U}}^{-}}\right\rangle.$ (86)

From Eqs. (100) and (105) with ${\alpha}=\ell,i,j$, we can obtain the mass
spectrum of the present model which includes $2N_{\rm t}$ real scalars and
$N_{\rm t}-1$ Weyl fermions with masses
$\widehat{m}_{\tilde{z}{\alpha}}=m_{3/2}$ where ${\alpha}=1,...,N_{\rm t}-1$.
The masses squared of the $2N_{\rm f}$ scalars are given in the
$C^{-}\rightarrow 0$ limit by Eqs. (38a) and (38b) for $C_{F}^{+}/C_{F}^{-}=1$
and $m_{3/2F}$ replaced by $m_{3/2{\cal FU}}$. In the same limit the masses
squared of the $2N_{\boldsymbol{\pm}}$ scalars are given by Eq. (75a) – (75d)
for $C_{\mathcal{U}}^{+}/C_{\mathcal{U}}^{-}=1$ and $m_{3/2{\cal U}}$ replaced
by $m_{3/2{\cal FU}}$.

To provide a pictorial verification of our present setting, we demonstrate in
Fig. 5 the three-dimensional plot of $V_{N\rm f\boldsymbol{\pm}}$ with $N_{\rm
f}=N_{\boldsymbol{-}}=1$ and $N_{\boldsymbol{+}}=0$, i.e. $V_{1\rm
f\boldsymbol{-}}$, versus $z_{1}$ and $z_{2}$ for
${\bar{z}}_{1}={\bar{z}}_{2}=0$ – see Eq. (17) – and the other parameters
arranged in column F of Table 1. Note that the subscripts $1$ and $2$ of $z$
correspond to $\ell=1$ and $j=1$ and the validity of Eqs. (77) and (82) is
protected. It is evident that the ground state, depicted by a tick black point
is totally stable. Some characteristic values of the masses of the relevant
particles are also arranged in column F of Table 1.

Fig. 5: The (dimensionless) SUGRA potential $V_{\rm
f\boldsymbol{-}}/m^{2}m_{\rm P}^{2}$ in Eq. (85) as a function of $z_{1}$ and
$z_{2}$ for ${\bar{z}}_{1}={\bar{z}}_{2}=0$ – see Eq. (17) – and the inputs
shown in column F of Table 1. The location of the dS vacuum in Eq. (26b) is
also depicted by a thick black point.

##  VI Link to the Observable Sector

Our next task is to study the transmission of the SUSY breaking to the visible
world. Here we restrict for simplicity ourselves to the cases with just one
Goldstino superfield, $Z$. To implement our analysis, we introduce the chiral
superfields of the observable sector $\Phi_{\alpha}$ with ${\alpha}=1,...,5$
and assume the following structure – cf. Ref. nilles ; susyr ; soft  – for the
total superpotential, $W_{\rm HO}$, of the theory

$W_{\rm HO}=W_{\rm H}(Z)+W_{\rm O}\left(\Phi_{\alpha}\right),$ (87)

where $W_{\rm H}$ is given by Eq. (14) or Eq. (49) for flat or curved $Z$
geometry respectively whereas $W_{\rm O}$ has the following generic form

$W_{\rm O}=h\Phi_{1}\Phi_{2}\Phi_{3}+\mu\Phi_{4}\Phi_{5}.$ (88)

with $h$ and $\mu$ free parameters. On the other hand, we consider three
variants of the total $K$ of the theory, $K_{\rm HO}$, ensuring universal SSB
parameters for $\Phi_{\alpha}$:

$\displaystyle K_{1\rm HO}$ $\displaystyle=K_{\rm
H}(Z)+\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2};~{}~{}~{}$ (89a)
$\displaystyle K_{2\rm HO}$ $\displaystyle=K_{\rm H}(Z)+N_{\rm
O}\ln\left(1+\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2}/N_{\rm O}\right),$
(89b) where $K_{\rm H}(Z)$ may be identified with $K_{\rm f}$ in Eq. (8) or
$K_{\pm}$ in Eq. (41) for flat or curved $Z$ geometry respectively whereas
$N_{\rm O}$ may remain unspecified. For curved $Z$ geometry we may introduce
one more variant $K_{3\rm HO}=\pm
n\ln\Big{(}\Omega_{\boldsymbol{\pm}}\pm\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2}/n\Big{)}.$
(89c)

If we expand the $K_{\rm HO}$’s above for low $\Phi_{\alpha}$ values, these
may assume the form

$K_{\rm HO}=K_{\rm H}(Z)+\widetilde{K}_{\rm H}(Z)|\Phi_{\alpha}|^{2},$ (90a)
with $\widetilde{K}_{\rm H}$ being identified as $\widetilde{K}_{\rm
H}=\begin{cases}1&\mbox{for}~{}~{}K_{\rm HO}=K_{1\rm HO},K_{2\rm HO};\\\
\Omega_{\boldsymbol{\pm}}^{-1}&\mbox{for}~{}~{}K_{\rm HO}=K_{3\rm
HO}\,.\end{cases}$ (90b)

Adapting the general formulae of Ref. soft ; susyr  to the case with one
hidden-sector field and tiny $\left\langle{V}\right\rangle$, we obtain the SSB
terms in the effective low energy potential which can be written as

$V_{\rm
SSB}=m_{\alpha}^{2}|\widehat{}\Phi_{\alpha}|^{2}+\left(A\widehat{h}\widehat{}\Phi_{1}\widehat{}\Phi_{2}\widehat{}\Phi_{3}+B\widehat{\mu}\widehat{}\Phi_{4}\widehat{}\Phi_{5}+{\rm
h.c.}\right),$ (91)

where the rescaled parameters are

$\widehat{h}=e^{\langle{K_{\rm H}}\rangle/2}\langle{\widetilde{K}_{\rm
H}}\rangle^{-3/2}h~{}~{}\mbox{and}~{}~{}\widehat{\mu}=e^{\langle{K_{\rm
H}}\rangle/2}\langle{\widetilde{K}_{\rm H}}\rangle^{-1}\mu$ (92)

and the canonically normalized fields are
$\widehat{}\Phi_{\alpha}=\langle{\widetilde{K}_{\rm
H}}\rangle^{1/2}\Phi_{\alpha}$.

In deriving the values of the SSB parameters above, we distinguish the cases:

##### (a)

For flat $Z$ geometry, i.e. $K_{\rm H}=K_{\rm f}$, we see from Eq. (90b) that
$\widetilde{K}_{\rm H}$ is constant for both adopted $K_{\rm HO}$’s and so,
the results are common. Substituting

$\left\langle{F^{Z}}\right\rangle=\sqrt{3}m_{3/2f}~{}~{}\mbox{and}~{}~{}\left\langle{\partial_{Z}K_{\rm
H}}\right\rangle={\rm v},$ (93)

into the relevant expressions susyr  we arrive at

$\displaystyle m_{\alpha}$ $\displaystyle=\left(1+e^{\sqrt{6}{\rm
v}}{V_{\Lambda}}/{m^{2}\left\langle{C_{f}^{-}}\right\rangle^{2}}\right)m_{3/2f}\simeq
m_{3/2f},$ (94) $\displaystyle A$ $\displaystyle=-\sqrt{\frac{3}{2}}{\rm
v}\left\langle{\frac{C_{f}^{+}}{C_{f}^{-}}}\right\rangle
m_{3/2f}\simeq\sqrt{\frac{3}{2}}{\rm v}m_{3/2f},$ $\displaystyle B$
$\displaystyle=A-m_{3/2f},$

where $(\widehat{h},\widehat{\mu})=e^{{\rm v}^{2}/4}(h,\mu)$ and the last
simplified expressions are obtained in the realistic limit $C^{-}\rightarrow
0$ which implies $C_{f}^{+}/C_{f}^{-}\rightarrow-1$.

##### (b)

For curved $Z$ geometry, i.e. $K_{\rm H}=K_{\boldsymbol{\pm}}$, we can
distinguish two subcases depending on which $K_{\rm HO}$ from those shown in
Eq. (89a) – (89c) is selected. Namely,

  * •

If $K_{\rm HO}=K_{1\rm HO}$ or $K_{2\rm HO}$, then $\widetilde{K}_{\rm H}$ in
Eq. (90b) is $Z$ independent. For $K_{\rm H}=K_{\boldsymbol{\pm}}$
respectively we find

$\displaystyle(m_{\alpha},A,B)=\left(1,\sqrt{3/2}{\rm
v},A/m_{3/2\boldsymbol{\pm}}-1\right)m_{3/2\boldsymbol{\pm}},$ (95)

where
$(\widehat{h},\widehat{\mu})=\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{\pm
n/2}(h,\mu)$ and we take into account the following

$\left\langle{F^{Z}}\right\rangle=\sqrt{3}\left\langle{v_{\boldsymbol{\pm}}}\right\rangle
m_{3/2\boldsymbol{\pm}}~{}~{}\mbox{and}~{}~{}~{}\left\langle{\partial_{Z}K_{\boldsymbol{\pm}}}\right\rangle=\left\langle{Zv_{\boldsymbol{\pm}}^{-1}}\right\rangle.$

  * •

If $K_{\rm HO}=K_{3\rm HO}$, then $\widetilde{K}_{\rm H}$ in Eq. (90b) is $Z$
dependent. Inserting the expressions

$\left\langle{\partial_{Z}\ln\widetilde{K}_{\rm
H}}\right\rangle=\frac{1}{n\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{2}}~{}~{}\mbox{and}\left\langle{\partial_{Z}\ln\widetilde{K}_{\rm
H}^{2}}\right\rangle=\frac{2{\rm
v}}{n\left\langle{v_{\boldsymbol{\pm}}}\right\rangle}$

into the general formulae soft ; susyr  we end up with the following results
for $K_{\rm H}=K_{\boldsymbol{\pm}}$ correspondingly:

$\displaystyle m_{\alpha}$ $\displaystyle=\sqrt{1\pm
3/n}\,m_{3/2\boldsymbol{\pm}};$ (96) $\displaystyle A$
$\displaystyle=\sqrt{3/2}{\rm v}(1\pm 3/n)m_{3/2\boldsymbol{\pm}};$
$\displaystyle B$ $\displaystyle=\left(\sqrt{3/2}{\rm v}(1\pm
2/n)-1\right)m_{3/2\boldsymbol{\pm}},$

where $\widehat{h}=\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{(3\pm
n)/2}h$ and
$\widehat{\mu}=\left\langle{v_{\boldsymbol{\pm}}}\right\rangle^{(2\pm
n)/2}\mu$.

In both cases above we take $C^{-}\simeq 0$ for simplicity. Note that the
condition $n>3$ for $K=K_{\boldsymbol{-}}$ which is imperative for the
stability of the configurations in Eqs. (10a) and (10b) – see Eq. (51b) –
implies non-vanishing SSB parameters too. Taking advantage from the numerical
inputs listed in columns A, C and D of Table 1 (for the three unimodular
models) we can obtain some explicit values for the SSB parameters derived
above – restoring units for convenience. Our outputs are arranged in the three
rightmost columns of Table 2 for the specific forms of $W_{\rm H},K_{\rm H}$
and $K_{\rm HO}$ in Eqs. (87) and (90a) shown in the three leftmost columns.
We remark that there is a variation of the achieved values of SSB parameters
which remain of the order of the $\widetilde{G}$ mass in all cases.

Table 2: SSB Parameters: a Case Study for the Inputs of Table 1. Input Settings | SSB Parameters in GeV
---|---
$W_{\rm H}$ | $K_{\rm H}$ | $\widetilde{K}_{\rm H}$ |  $m_{\alpha}$ | $|A|$ | $|B|$
$W_{\Lambda\rm f}$ | $K_{\rm f}$ | $1$ |  $170$ | $208$ | $378$
$W_{\Lambda\boldsymbol{+}}$ | $K_{\rm\boldsymbol{+}}$ | $1$ |  $145$ | $177$ | $32$
$W_{\Lambda\boldsymbol{+}}$ | $K_{\rm\boldsymbol{+}}$ | $1/\Omega_{\boldsymbol{+}}$ |  $290$ | $711$ | $388$
$W_{\Lambda\boldsymbol{-}}$ | $K_{\rm\boldsymbol{-}}$ | $1$ |  $179$ | $220$ | $40$
$W_{\Lambda\boldsymbol{-}}$ | $K_{\rm\boldsymbol{-}}$ | $1/\Omega_{\boldsymbol{-}}$ |  $90$ | $55$ | $69$

##  VII Conclusions

We have extended the approach of Ref. noscale18 ; noscale19 , proposing new
no-scale SUGRA models which lead to Minkowski, dS and AdS vacua without need
for any external uplifting mechanism. We first provided a simple but general
enough argument which assists to appreciate the effectiveness of our paradigm.
We then adopted specific single-field models and showed that the achievement
of dS/AdS solutions using pairs of Minkowski ones works perfectly well for
flat – see Eqs. (8) and (14) – and hyperbolic or spherical geometry – see Eqs.
(41) and (49). We also broadened these constructions to multi-field models –
see Sec. B, B and V. Within each case we derived the SUGRA potential and the
relevant mass spectrum paying special attention to the stability of the
proposed solutions. Typical representatives of our results were illustrated in
Fig. 1 – 5 employing numerical inputs from Table 1. We provided, finally, the
set of the soft SUSY-breaking parameters induced by our unimodular models
linking them to a generic observable sector – see Eqs. (94) - (96). We
verified – see Table 2 – that their magnitude is of the order of the gravitino
mass.

As stressed in Ref. noscale19 ; noscaleinfl , this kind of constructions,
based exclusively in SUGRA, can be considered as part of an effective theory
valid below $m_{\rm P}$. However, the correspondence between Kähler and super-
potentials which yields naturally Minkowski, dS and AdS (locally stable) vacua
with broken SUSY may be a very helpful guide for string theory so as to
establish new possible models with viable low energy phenomenology. As regards
the ultraviolet completion, it would be interesting to investigate if our
models belong to the string landscape or swampland vafa . Note that the
swampland string conjectures are generically not satisfied in SUGRA-based
models but there are suggestions ferara ; sevilla  which may work in our
framework too. One more open issue is the interface of our settings with
inflation. We aspire to return on this topic soon taking advantage from other
similar studies noscaleinfl ; ketov ; lhclinde ; ant  – see Ref. deinf . At
last but not least, let us mention that the achievement of the present value
of the dark-energy density parameter in Eq. (7) requires an inelegant fine
tuning, which may be somehow alleviated if we take into account contributions
from the electroweak symmetry breaking and/or the confinement in quantum
chromodynamics noscale19 ; noscaleinfl .

Despite the shortcomings above, we believe that the establishment of novel
models for SUSY breaking with a natural emergence of Minkowski and dS/AdS
vacua can be considered as an important development which offers the
opportunity for further explorations towards several cosmo-phenomenological
directions.

#### Acknowledgment

This research work was supported by the Hellenic Foundation for Research and
Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to
support Faculty members and Researchers and the procurement of high-cost
research equipment grant” (Project Number: 2251).

#### Dedication

I would like to dedicate the present paper to the memory of T. Tomaras, an
excellent University teacher who let his imprint on my first post-graduate
steps.

##  Appendix A Mass Formulae In SUGRA

We here generalize our formulae in Ref. susyr  for $N$ chiral multiplets and
dS/AdS vacua. Let us initially remind that central role in the SUGRA formalism
plays the Kähler-invariant function expressed in terms of the Kähler potential
$K$ and the superpotential $W$ as follows

$G:=K+\ln|W|^{2}.$ (97)

Using it we can derive the F-term scalar potential nilles

$V=e^{G}\left(G^{{\alpha}{\bar{\beta}}}G_{\alpha}G_{\bar{\beta}}-3\right)=e^{K}|W|^{2}\left(K^{{\alpha}{\bar{\beta}}}G_{\alpha}G_{\bar{\beta}}-3\right),$
(98)

where the subscripts of quantities $G$ and $K$ denote differentiation w.r.t
the superfields $Z_{\alpha}$ and
$G^{{\alpha}{\bar{\beta}}}=K^{{\alpha}{\bar{\beta}}}$ is the inverse of the
Kähler metric $K_{{\alpha}{\bar{\beta}}}$. The spontaneous SUSY breaking takes
place typically at a (locally stable) vacuum or flat direction of $V$ which
satisfies the extremum and minimum conditions

$\left\langle{\partial_{{\alpha}}V}\right\rangle=\left\langle{\partial_{\bar{\alpha}}V}\right\rangle=0~{}~{}\mbox{and}~{}~{}\widehat{m}_{A}^{2}>0.$
(99)

Here $\partial_{{\alpha}}:=\partial/\partial Z_{\alpha}$ and
$\partial_{\bar{\alpha}}:=\partial/\partial Z^{*}_{\bar{\alpha}}$ with the
scalar components of the superfields denoted by the same superfield symbol.
Also $\widehat{m}_{A}^{2}$ are the eigenvalues of the $2N\times 2N$ mass-
squared matrix $M^{2}_{0}$ of the (canonically normalized) scalar fields which
is computed applying the formula

$M^{2}_{0}=\mbox{$\left\lgroup\begin{matrix}\left\langle{\partial_{\widehat{\alpha}}\partial_{\widehat{\beta}}V}\right\rangle&\left\langle{\partial_{\widehat{\alpha}}\partial_{\widehat{\bar{\beta}}}V}\right\rangle\cr\left\langle{\partial_{\widehat{\bar{\beta}}}\partial_{\widehat{\alpha}}V}\right\rangle&\left\langle{\partial_{\widehat{\bar{\beta}}}\partial_{\widehat{\bar{\alpha}}}V}\right\rangle\end{matrix}\right\rgroup$},$
(100)

where $\partial_{\widehat{A}}:=\partial/\partial\widehat{Z}_{A}$ with
$A={\alpha}$ or $\bar{\alpha}$ and
$\widehat{Z}_{\alpha}=\sqrt{K_{{\alpha}{\bar{\alpha}}}}Z_{\alpha}$ given that
the $K$’s considered in our work are diagonal.

The aforementioned $M^{2}_{0}$ is one of the mass-squared matrices $M_{J}^{2}$
of the particles with spin $J$, composing the spectrum of the model. They obey
the super-trace formula nilles ; mass

$\displaystyle{\sf STr}M^{2}$
$\displaystyle:=\sum_{J=0}^{3/2}(-1)^{2J}(2J+1)\mbox{\sf Tr}M_{J}^{2}$ (101a)
$\displaystyle=2m_{3/2}^{2}\Big{(}\left\langle{(N-1)(1+V/m_{3/2}^{2})}\right\rangle$
$\displaystyle+\left\langle{G_{\alpha}G^{{\alpha}{\bar{\beta}}}R_{\beta\bar{\gamma}}G^{\gamma\bar{\delta}}G_{\bar{\delta}}}\right\rangle\Big{)},$
(101b)

where $R_{{\alpha}{\bar{\beta}}}$ is the (moduli-space) Ricci curvature which
reads

$R_{{\alpha}{\bar{\beta}}}=-\partial_{\alpha}\partial_{\bar{\beta}}\ln{\rm
det}\left(G_{\gamma\bar{\delta}}\right).$ (102)

Note that Eq. (101b) provides a geometric computation of ${\sf STr}M^{2}$
which can be employed as an consistency check for the correctness of a direct
computation via the extraction of the particle spectrum by applying Eq.
(101a).

The factor $N-1$ in the first term of Eq. (101b) reflects the fact that we
obtain one fermion with spin $1/2$ less than the number $N$ of the chiral
multiplets. This is because one such fermion, known as goldstino, is absorbed
by the gravitino ($\widetilde{G}$) with spin $3/2$ according to the super-
Higgs mechanism nilles . The $\widetilde{G}$ mass squared is evaluated as
follows

$m_{3/2}^{2}=\left\langle{e^{G}}\right\rangle=\frac{1}{3}\left\langle{G_{{\alpha}{\bar{\beta}}}F^{\alpha}F^{*{\bar{\beta}}}-V}\right\rangle,$
(103)

where the F terms are defined as soft

$F^{\alpha}:=e^{G/2}K^{{\alpha}{\bar{\beta}}}G_{\bar{\beta}}~{}~{}\mbox{and}~{}~{}F^{*{\bar{\alpha}}}:=e^{G/2}K^{{\bar{\alpha}}{\beta}}G_{\beta}\,.$
(104)

In our work we compute also the elements of $M_{1/2}$, i.e., the masses of the
(canonically normalized) chiral fermions, $\tilde{Z}_{\alpha},$ which can be
found applying the formula

$m_{{\alpha}{\beta}}=m_{3/2}\left\langle{G_{{\alpha}{\beta}}+(1-2/U)G_{{\alpha}}G_{{\beta}}}\right\rangle\left\langle{G_{{\alpha}{\bar{\alpha}}}G_{{\beta}{\bar{\beta}}}}\right\rangle^{-1/2},$
(105)

where $G_{{\alpha}{\beta}}$ is defined in terms of the Kähler-covariant
derivative $D_{\alpha}$ as

$G_{{\alpha}{\beta}}:=D_{\alpha}G_{\beta}=\partial_{\alpha}G_{{\beta}}-\Gamma^{\gamma}_{{\alpha}{\beta}}G_{\gamma},$
(106)

with
$\Gamma^{\gamma}_{{\alpha}{\beta}}=K^{\gamma\bar{\gamma}}\partial_{\alpha}K_{{\beta}\bar{\gamma}}$
and $U$ takes into account a possible non-vanishing
$\left\langle{V}\right\rangle$, i.e.,

$U=3+\left\langle{V}\right\rangle/m_{3/2}^{2}.$ (107)

In Eq. (105) care is taken so as to canonically normalize the various fields
and remove the mass mixing between $\widetilde{G}$ and fields with spin $1/2$
in the SUGRA lagrangian.

Let us, finally, note that Eq. (101b) can be significantly simplified for
$N=1$ since it can be brought into the form

$\displaystyle{\sf STr}M^{2}$
$\displaystyle=2m_{3/2}^{2}\langle{G_{Z{Z^{*}}}^{-2}G_{Z}G_{{Z^{*}}}R_{Z{Z^{*}}}}\rangle$
(108)
$\displaystyle=2m_{3/2}^{2}\left\langle{(3+V/m_{3/2}^{2}){\mathcal{R}}}\right\rangle,$

where we make use of Eq. (98) and the definition of the scalar curvature
$\mathcal{R}$ which is

$\mathcal{R}=G^{{\alpha}{\bar{\beta}}}R_{{\alpha}{\bar{\beta}}}.$ (109)

Note that the first term of Eq. (101b) vanishes for $N=1$ due to the super-
Higgs effect.

##  Appendix B Half-Plane Parametrization

In this Appendix we employ the half-plane parametrization of the hyperbolic
geometry which allows us to compare our results in Sec. A with those
established in Ref. noscale19 ; noscale18 . The transformation from the disc
coordinates $Z$ and $Z^{*}$, utilized in Sec. A, to the new ones $T$ and
$T^{*}$ is performed linde ; old ; su11  via the replacement

$Z=-\sqrt{n}\frac{T-1/2}{T+1/2}~{}~{}\mbox{with}~{}~{}{\sf Re}(T)>0.$ (110)

The last restriction – from which the name of the $T-T^{*}$ coordinates – is
compatible with Eq. (42) for $K=K_{\boldsymbol{-}}$.

Inserting Eq. (110) into Eqs. (41) and (46), $K_{\boldsymbol{-}}$ and
$W_{0\boldsymbol{-}}$ may be expressed in terms of $T$ and $T^{*}$ as follows

$\displaystyle K_{\boldsymbol{-}}$ $\displaystyle=$
$\displaystyle-n\ln\frac{T+T^{*}}{(T+1/2)(T^{*}+1/2)}$ (111a)
$\displaystyle\mbox{and}~{}~{}W_{0\boldsymbol{-}}^{\pm}$ $\displaystyle=$
$\displaystyle(2T)^{n_{\mp}}(T+1/2)^{-n},$ (111b)

where we fix $k=0$ in Eq. (41), define the exponents

$n_{\pm}=\frac{1}{2}\left(n\pm\sqrt{3n}\right)$ (112)

and take into account the identity

${\rm atnh}\frac{Z}{\sqrt{n}}=\frac{1}{2}\ln\frac{\sqrt{n}+Z}{\sqrt{n}-Z}.$
(113)

Performing a Kähler transformation as in Eq. (21) with

$\Lambda_{K}=-n\ln\left(T+1/2\right)$ (114)

we can show that the model described by Eqs. (111a) and (111b) is equivalent
to a model relied on the following ingredients

$\widetilde{K}_{\boldsymbol{-}}=-n\ln\left({T+\bar{T}}\right)~{}~{}\mbox{and}~{}~{}\widetilde{W}_{\boldsymbol{-}}^{\pm}=m(2T)^{n_{\mp}}.$
(115)

We reveal the celebrated $K$ and $W$ analyzed in Ref. noscale18 ; noscale19 .
Contrary to the solutions proposed in Eqs. (13) and (47), the presence of the
exponents in Eq. (115) may require some special attention from the point of
view of holomorphicity noscale18 ; noscale19 . Considering, though,
$\widetilde{W}_{\boldsymbol{-}}^{\pm}$ as an effective $W$, valid close to the
non-zero vacuum of the theory, any value of $n_{\pm}$ is, in principle,
acceptable.

Trying to achieve locally stable dS/AdS vacua with stabilized $T$ we
concentrate on the following $K$

$\widetilde{K}_{\boldsymbol{-}}=-n\ln\widetilde{\Omega},$ (116a) where the
argument of $\ln$ is introduced as $\widetilde{\Omega}=T+T^{*}+k^{2}T_{\rm
v}^{4}/n~{}~{}\mbox{with}~{}~{}T_{\rm v}=T+T^{*}-\sqrt{2}{\rm v}.$ (116b)

As regards $W$, this can be generated by interconnecting the two parts in Eq.
(115). Namely, we define

$\widetilde{W}_{\Lambda\boldsymbol{-}}=C^{+}\widetilde{W}_{0\boldsymbol{-}}^{+}-C^{-}\widetilde{W}_{0\boldsymbol{-}}^{-}=m(2T)^{n_{+}}C_{T}^{-},$
(117)

where the last short expression is achieved thanks to the new $C$ symbols
defined as

$C_{T}^{\pm}:=C^{+}\pm C^{-}(2T)^{-\sqrt{3n}}.$ (118)

The resulting SUGRA potential $\widetilde{V}_{\boldsymbol{-}}$, obtained after
replacing Eqs. (116a) and (117) into Eq. (98), is found to be

$\displaystyle\widetilde{V}_{\boldsymbol{-}}$
$\displaystyle=(m/2)^{2}\widetilde{\Omega}^{-n}|2T|^{2n_{+}}\Big{(}-12|C_{T}^{-}|^{2}+n\widetilde{\Omega}^{2}\cdot$
(119)
$\displaystyle\left|\left(\sqrt{3}C_{T}^{+}+\sqrt{n}C_{T}^{-}\right)/T-2\sqrt{n}(1+4k^{2}T_{\rm
v}^{3}/n)C_{T}^{-}\widetilde{\Omega}^{-1}\right|^{2}$
$\displaystyle\cdot\left(n\left(1+4k^{2}T_{\rm
v}^{3}/n\right)^{2}-12k^{2}T_{\rm
v}^{2}\widetilde{\Omega}\right)^{-1}\Big{)}.$

For the directions in Eqs. (10a) and (10b) – with $Z$ replaced by $T$ – we
obtain dS/AdS vacua since
$\langle{\widetilde{V}_{\boldsymbol{-}}}\rangle=V_{\Lambda}$ given in Eq. (6).
In addition, the conditions in Eq. (99) for $V=\widetilde{V}_{\boldsymbol{-}}$
are satisfied after imposing $n>3$. This is, because the sgoldstino components
($t$ and $\bar{t}$) – appearing by the decomposition of $T$ as in Eq. (17) –
acquire masses squared

$\displaystyle\widehat{m}_{t}^{2}$ $\displaystyle=$ $\displaystyle
288\sqrt{2}k^{2}n^{-1}{\rm
v}^{3}\left\langle{{C_{T}^{+}}/{C_{T}^{-}}}\right\rangle^{2}\widetilde{m}_{3/2}^{2};$
(120a) $\displaystyle\widehat{m}_{\bar{t}}^{2}$ $\displaystyle=$
$\displaystyle
4\left(1-(3/n)\left\langle{C_{T}^{+}/C_{T}^{-}}\right\rangle^{2}\right)\widetilde{m}_{3/2}^{2}.$
(120b)

Note that the expression for $\widehat{m}_{\bar{t}}$ coincides with that for
$\widehat{m}_{\bar{z}}$ in Eq. (51b) for $K=K_{\boldsymbol{-}}$ if we replace
$C_{T}^{\pm}$ with $C_{u\boldsymbol{-}}^{\pm}$. As in that case, to ensure
$\widehat{m}_{\bar{t}}^{2}>0$ we have to impose the aforementioned lower bound
on $n$. Otherwise, an extra term of the form $\tilde{k}(T-T^{*})^{4}$
noscale18 ; noscale19  added in Eq. (116b) may facilitate the stabilization
for lower $n$ values. The expressions above contain the $\widetilde{G}$ mass

$\widetilde{m}_{3/2}=m(\sqrt{2}{\rm
v})^{\sqrt{3n}/2}\left\langle{C_{T}^{-}}\right\rangle$ (121)

which can be determined after inserting Eqs. (116a) and (117) into Eq. (103).
Upon substitution of the the mass spectrum above into Eq. (101a) we find

${\sf
STr}M_{T\boldsymbol{-}}^{2}=(12/n)\widetilde{m}_{3/2}^{2}\left\langle{C_{T}^{+}/C_{T}^{-}}\right\rangle^{2}\left(24\sqrt{2}k^{2}{\rm
v}^{3}/n-1\right),$ (122)

consistently with the expression of Eq. (108) given that $\mathcal{R}$ from
Eq. (109) is

$\left\langle{\widetilde{\cal
R}_{T\boldsymbol{-}}}\right\rangle=\left(24\sqrt{2}k^{2}{\rm
v}^{3}/n-1\right)/{\rm v}^{2}.$ (123)

Adopting the superpotential in Eq. (88) for the visible-sector fields
$\Phi_{a}$ and employing for simplicity $C^{-}\simeq 0$ we below find the
resulting SSB parameters. To this end, we identify $K_{\rm H}$ in Eqs. (89a)
and (89b) with $\widetilde{K}_{\boldsymbol{-}}$ in Eq. (116a) and so we obtain
the corresponding $\widetilde{K}_{1\rm HO}$ and $\widetilde{K}_{2\rm HO}$. On
the other hand, $K_{3\rm HO}$ in Eq. (89c) may be replaced with the following

$\widetilde{K}_{3\rm
HO}=-n\ln\left(\widetilde{\Omega}-\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2}/n\right).$
(124)

For low $\Phi_{\alpha}$ values, the $K_{\rm HO}$’s above reduce to that shown
in Eq. (90a), with $\widetilde{K}_{\rm H}$ being identified as

$\widetilde{K}_{\rm H}=\begin{cases}1&\mbox{for}~{}~{}K_{\rm
HO}=\widetilde{K}_{1\rm HO},\widetilde{K}_{2\rm HO};\\\
\widetilde{\Omega}^{-1}&\mbox{for}~{}~{}K_{\rm HO}=\widetilde{K}_{3\rm
HO}\,.\end{cases}$ (125)

Using the standard formalism susyr , we extract the following SSB masses
squared

$m_{\alpha}^{2}=\begin{cases}\widetilde{m}_{3/2}^{2}&\mbox{for}~{}~{}K_{\rm
HO}=\widetilde{K}_{1\rm HO},\widetilde{K}_{2\rm HO};\\\
(1-3/n)\widetilde{m}_{3/2}^{2}&\mbox{for}~{}~{}K_{\rm HO}=\widetilde{K}_{3\rm
HO},\end{cases}$ (126a) trilinear coupling constant
$\frac{A}{\widetilde{m}_{3/2}}=\begin{cases}-\sqrt{3n}&\mbox{for}~{}~{}K_{\rm
HO}=\widetilde{K}_{1\rm HO}\\\ &\mbox{and}~{}~{}\widetilde{K}_{2\rm HO};\\\
\sqrt{3}\left({3}/{\sqrt{n}}-\sqrt{n}\right)&\mbox{for}~{}~{}K_{\rm
HO}=\widetilde{K}_{3\rm HO},\\\ \end{cases}$ (126b) and bilinear coupling
constant
$\frac{B}{\widetilde{m}_{3/2}}=\begin{cases}-\left(1+\sqrt{3n}\right)&\mbox{for}~{}~{}K_{\rm
HO}=\widetilde{K}_{1\rm HO}\\\ &\mbox{and}~{}~{}\widetilde{K}_{2\rm HO};\\\
\left(\sqrt{3/n}(2-n)-1\right)&\mbox{for}~{}~{}K_{\rm HO}=\widetilde{K}_{3\rm
HO}.\end{cases}$ (126c)

To reach the results above we take into account the auxiliary expressions

$\displaystyle\left\langle{F^{T}}\right\rangle$
$\displaystyle=2\sqrt{3}\widetilde{m}_{3/2}/\sqrt{n},~{}\left\langle{\partial_{T}K_{\rm
H}}\right\rangle=-n/\sqrt{2}{\rm v}$ (127)
$\displaystyle\left\langle{\partial_{T}\ln\widetilde{K}_{\rm
H}^{3}}\right\rangle$
$\displaystyle={3}\left\langle{\partial_{T}\ln\widetilde{K}_{\rm
H}^{2}}\right\rangle/2=-3/\sqrt{2}{\rm v}$

and define the rescaled parameters

$\displaystyle(\widehat{h},\widehat{\mu})=\left(\sqrt{2}{\rm
v}\right)^{-n/2}(h,\mu)$

for $K_{\rm HO}=\widetilde{K}_{1\rm HO}$ and $\widetilde{K}_{2\rm HO}$. For
$K_{\rm HO}=\widetilde{K}_{3\rm HO}$ we have

$\displaystyle\widehat{h}=\left(\sqrt{2}{\rm
v}\right)^{(3-n)/2}h~{}~{}\mbox{and}~{}~{}\widehat{\mu}=\left(\sqrt{2}{\rm
v}\right)^{(2-n)/2}\mu\,.$

For $K_{\rm HO}=\widetilde{K}_{3\rm HO}$ and $n=3$ we recover the standard no-
scale SSB terms as regards $m_{\alpha}$ and $A$ old ; nsreview  but not for
$B$ – cf. Ref. noscaleinfl . The reason is that here $W$ in Eq. (117) is not
constant as in the original no-scale models and this fact modifies the
resulting $\left\langle{F^{T}}\right\rangle$ which includes derivation of $W$
w.r.t $T$. Comparing the above results with those in Eqs. (95) and (96) we
remark that the expressions for $m_{\alpha}$ are exactly the same.

Extensions of the present model including more than one goldstini and also
matter fields are extensively investigated in Ref. noscale18 ; noscale19 .

## References

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