KIAS-P21017

Swampland de Sitter Conjectures in
No-Scale Supergravity Models

Ida M. Rasulian1, Mahdi Torabian1, and Liliana Velasco-Sevilla2,3

1Department of Physics, Sharif University of Technology, Tehran 11155-9161,
Iran;

2 Department of Physics and Technology, University of Bergen,
PO Box 7803, 5020 Bergen, Norway;

3Korea Institute for Advanced Study, Seoul 02455, Korea

###### Abstract

It is challenging to construct explicit and controllable models that realize
de Sitter solutions in string compactifications. This difficulty is the main
motivation for the _Refined de Sitter Conjecture_ and the _Trans-Planckian
Censorship Conjecture_ which forbid stable de Sitter solutions but allow
metastable, unstable and rolling solutions in a theory consistent with quantum
gravity. Inspired by this, we first study a toy de Sitter No-Scale
Supergravity model and show that for particular choices of parameters it can
be consistent with the Refined de Sitter Conjecture and the Trans-Planckian
Censorship Conjecture. Then we modify the model by adding rolling dynamics and
show that the theory can become stable along the imaginary direction, where it
would otherwise be unstable. We extend the model to multi-field rolling and de
Sitter fields, finding the parameter space where they can be compatible with
the Refined de Sitter Conjecture . The modified models with rolling fields can
be used to construct Quintessence models to accommodate the accelerating
expansion of the Universe.

###### Contents

  1. 1 Introduction and Motivation
  2. 2 No-Scale Models with Unstable dS Vacua
    1. No-Scale de Sitter Models
    2. Refined de Sitter Conjecture in No-Scale Models
    3. Trans-Planckian Conjecture in No-Scale de Sitter Models
  3. 3 No-Scale Models with Rolling Dynamics
    1. The Minimal Model
      1. Existence of Extrema
      2. Stability Condition
      3. Assertion of the Refined de Sitter Conjecture
      4. Superpotential with two Terms in the Expansion of the Rolling Field
  4. 4 Non-minimal No-Scale Models
  5. 5 Application to Quintessence Models
  6. 6 Conclusions
  7. A Canonical Normalization
  8. B Details of Stability Conditions

##  1 Introduction and Motivation

In recent years, the search for de Sitter (dS) solutions in superstring and
supergravity theories has intensified 111The literature is vast, see for
example [1, 2, 3, 4, 5, 6, 7, 8, 9]. due to two pivotal observations. The
first one is the discovery of the accelerating expansion of the Universe due
to non-vanishing vacuum energy [10, 11]. The second one is the observational
support for inflationary cosmology [12]. According to the latter observation,
it appears that the Universe underwent an early epoch of near-exponential
quasi-de Sitter expansion driven by an inflaton field energy that was large in
comparison with the electroweak scale of the Standard Model, but still
hierarchically smaller than the Planck scale.

Nevertheless, it has been challenging to construct explicit and controllable
models that realize de Sitter solutions in string compactifications [13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. This difficulty is
the main motivation for the proposal of the Refined de Sitter Conjecture
(RdSC), [29, 30] (see also [31]). In particular, the RdSC constrains the
scalar potential of the low energy effective field theory (EFT) and rules out
the existence of dS minima, nevertheless, dS maxima are allowed. Namely, the
scalar potential $V$ of a consistent EFT must satisfy either

$\displaystyle|\nabla V|\geq\frac{c}{M_{\rm{P}}}\ V\,,$ (1)

or

$\displaystyle\text{min}\left(\nabla_{i}\nabla_{j}V\right)\leq-\frac{c^{\prime}}{M^{2}_{\rm{P}}}\
V\,,$ (2)

where the constants $c$, $c^{\prime}$ are order one positive constants. In our
work we refer respectively to Eqs. (1) and (2) as the first and second de
Sitter criteria. They are referred with that terminology in the literature or
also with the name _conjectures_. Eqs. (1) and (2) imply that stable de Sitter
vacua do not exist, however, unstable vacua or rolling dS solutions can exist.
On the other hand, the Trans-Planckian Censorship Conjecture (TCC) [32],
another swampland conjecture related to dS background, puts an upper bound on
the lifetime, $\tau$, of a dS solution. In particular, besides unstable and
rolling dS solutions, it allows for metastable dS minima with limited
lifetime. If we call $H_{f}$, the Hubble rate at the end of dS phase, given by
$H_{f}\sim\sqrt{V/3\,M_{\rm{P}}^{2}}$, then the TCC implies

$\tau<\frac{1}{H_{f}}\,\ln\frac{M_{\rm{P}}}{H_{f}}.$ (3)

Hereafter we work in Planck units, hence $M_{\rm{P}}=1$. If the above bound,
Eq. (3), is applied to the scalar potential, $V$, then dS extrema are
forbidden in asymptotic regions of moduli space; however similar to the first
dS criteria, Eq. (1), rolling dS solutions are allowed for a fixed value of
$c=\sqrt{2/3}$. On the other hand, in the bulk of moduli space, the TCC allows
dS critical points (both minima and dS maxima) as long as their lifetime is
bounded.

Some studies indicate that within the Swampland Programme there is a web of
conjectures instead of a list of conjectures, i.e., the conjectures are
related to each other (see for instance [33]). As already pointed out, the TCC
implies the RdSC in the asymptotic region of the moduli space. Besides, the
TCC fixes the universal order one constant of the RdSC bound of Eq. (1).
Furthermore, there is an interesting coincidence between the TCC lifetime of a
dS solution and the scrambling time of a black hole [32]. The upper bound on
the lifetime, indicates that the dS vacuum is not a thermal state. Namely,
there is not enough time for the perturbations to get thermalized with the
state. The TCC condition can also be seen as a bound on the growth rate of the
entropy [34, 35, 36]. The latter in particular indicates a deep connection
between the TCC and a principle of quantum gravity. We recall that the dS
state is a state with finite entropy and, depending on its natural scale (its
Hubble rate), there is a bound on how fast it can be saturated with
microstates. There are indications [37] that the UV/IR decoupling limit of
effective field theories (without gravity) might not be working in the
presence of gravity. In this sense, the TCC bound offers a way to test the IR
cutoff (the Hubble scale) and the UV cutoff (the Planck scale) of a theory.
This is because the TCC is a concrete example of the UV/IR non-decoupling
effects, hence offering a way to understand a theory in terms of a principle
of quantum gravity. All these arguments lead us to apply the TCC besides the
RdSC in our present bottom-up study.

Although a priori there is no no-go theorem, it is not clear whether string
theory admits dS solutions. The RdSC belongs to the swampland programme that
aims at determining the list of criteria an EFT must admit to be embedded in a
theory of quantum gravity. Namely, not every quantum field theory admits an
ultraviolet completion when quantum gravity is considered. For bottom-up model
building, the list of swampland conjectures helps to construct consistent
models that can possibly descend from string theory.

In [38], and then in [39] for particular cases, de Sitter no-scale models have
been constructed by building de Sitter plateaus from the combination of
Minkowski solutions. These scenarios have been then subsequently and
extensively applied to construct inflation models, [40, 41, 42, 43]. No-scale
supergravity models avoid technical problems of generic supergravity theories
and thus are useful for making realistic calculable models. One may wonder if
no-scale supergravity models which have dS vacua admit the swampland
conjectures and thus can be derived from strings or not, in which case
belonging to the swampland. Recently in [44] the RdSC were re-phrased in the
context of ${\cal N}=1$ supergravity chiral multiplets. In particular for a
positive potential $V>0$ the first criterion of RdSC, Eq. (1), implies

$\frac{K^{I\bar{J}}\,\partial_{I}V\,\partial_{\bar{J}}\,V}{V^{2}}\geq
c^{2}/2,$ (4)

where $K^{I\bar{J}}$ is the inverse of the Kähler metric and $I$ runs over all
scalar components of chiral superfields. In this paper we use the proposal of
[44], which offers an escape to Eq. (4), for models with de Sitter vacua by
modifying the theory with the addition of rolling dynamics. This proposal adds
an exponential dependence on the rolling field to the original scalar
potential, $V_{0}$, such that the full scalar potential becomes
$\widehat{V}=e^{\lambda\,\chi}\,V_{0}$, with $\lambda$ constant. Then
$\widehat{V}$ can be consistent with Eq. (4) (with the appropriate replacement
$V\rightarrow\widehat{V}$) for some choice of parameters. Using the above
approach, we find that the modification of adding a rolling dynamics, opens up
the region where the models survive the swamp. We also find that for some
cases these theories can be used for Quintessence [45].

The paper is organized as follows: in section 2 we present the statement of
the Refined de Sitter Conjecture in the context of [44], then we identify the
necessary conditions for the $\alpha$\- supergravity/no-scale supergravity
theories to satisfy Eq. (2) and the Trans-Planckian Conjecture, Eq. (3) . In
section 3 we study how adding an exponential factor to the
$\alpha$-supergravity/no-scale models widens the parameter space to be
compatible with Eqs. (1) and ,(2). In section 4 we study multifield models,
containing more than one rolling field or more than one de Sitter field. In
section 5 we talk about how the models we present can be used as Quintessence
models. In section 6 we conclude.

##  2 No-Scale Models with Unstable dS Vacua

### No-Scale de Sitter Models

Stable supergravity theories parameterizing a symmetric geometry [46, 47] have
provided the basis to construct a broad class of superconformal inflationary
models with a universal observational prediction for the spectral index of
$n_{s}=1-2/N$ and $r=12/N^{2}$, as favored by the Planck data [48], where $N$
is the $N$-folds necessary for inflation. This class of models has been
generalized through the introduction of the parameter $\alpha$, inversely
proportional to the curvature of the inflaton Kähler manifold in the limit of
sufficiently large curvature ($\alpha$ small). As mentioned in the
introduction, a construction of this kind of models as a deformation of no-
scale models was constructed by building de Sitter plateaus from the
combination of Minkowski solutions [38, 39]. These models have been then
further developed and analyzed in [40, 41, 42, 43] using the following Kähler
potential and superpotential

$\displaystyle K$ $\displaystyle=$
$\displaystyle-3\,\sum_{i=1}^{N}\,\alpha_{i}\,\ln(\phi_{i}+\bar{\phi}_{i}),$
(5) $\displaystyle W$ $\displaystyle=$ $\displaystyle
a\,\left(\prod_{i=1}^{N}\,\phi_{i}^{n_{i+}}-\prod_{i=1}^{N}\,\phi_{i}^{n_{i-}}\right),$
(6)

where $i$ runs over $N$ no-scale chiral superfields, $\alpha_{i}>0$, $a$ is an
arbitrary constant and

$n_{i\,\pm}=\frac{3}{2}\left(\alpha_{i}\pm\frac{r_{i}}{s}\right)\quad{\rm
for}\quad{\textstyle\sum_{i=1}^{N}\,r_{i}^{2}=1},\quad
s^{2}={\textstyle\sum_{j=1}^{N}\,\frac{r_{j}^{2}}{\alpha_{j}}}.$ (7)

The scalar potential

$\displaystyle V=e^{G}(X-3),\quad
X=G_{i}\,K^{i\bar{\jmath}}\,G_{\bar{\jmath}},$ (8)

where $K^{i\bar{\jmath}}$ is the inverse Kähler metric and $G=K+\ln|W|^{2}$;
has an extremum, $\nabla V=0$, along the real direction (i.e.
$\phi_{i}=\bar{\phi}_{i}$) and becomes a constant in that direction [40]

$\displaystyle V=3\,a^{2}\,2^{2-3\sum_{i=1}^{N}\,\alpha_{i}}.$ (9)

In [40] it was noted that in order to render the theory stable a quartic term
could be added to the Kähler potential in the following way

$\displaystyle
K=-3\,\sum_{i}^{3}\,\alpha_{i}\,\ln\left[\phi_{i}+\overline{\phi}_{i}+b_{i}(\phi_{i}-\overline{\phi}_{i})^{4}\right],$
(10)

such that the imaginary direction is stabilized and hence the full potential
(that is when taking into account real and imaginary terms) can remain de
Sitter and hence bounded from below.

### Refined de Sitter Conjecture in No-Scale Models

The potential of Eq. (9) is positive and hence it is a dS solution, being
constant along the real direction, cannot satisfy the first criterion of RdSC,
Eq. (1), since the constant $c$ appearing there cannot be zero. In the
following, we study a simple single field no-scale model and show that the
instability along the imaginary direction makes the model compatible with the
second criterion of RdSC (2). The Kähler potential and the superpotential of
the model are

$\displaystyle K$ $\displaystyle=$
$\displaystyle-3\,\alpha\ln(\phi+\bar{\phi}),$ $\displaystyle W$
$\displaystyle=$ $\displaystyle a\,(\phi^{n_{+}}-\phi^{n_{-}}),$ (11)

where $\alpha>0$ and $n_{\pm}=3/2(\alpha\pm\sqrt{\alpha})$. The scalar
potential is

$\displaystyle V$ $\displaystyle=$ $\displaystyle a^{2}\
\left(\phi+\bar{\phi}\right)^{-3\,\alpha}\,|\phi^{n_{+}}-\phi^{n-}|^{2}\left(X-3\right),$
(12)

where

$\displaystyle X=G_{i}\,K^{i\bar{\jmath}}\,G_{\bar{\jmath}}=$ (13)
$\displaystyle\frac{\left[-(\phi^{n_{-}}-\phi^{n_{+}})+\frac{(\phi+\bar{\phi})}{3\,\alpha}\,(n_{-}\phi^{n_{-}-1}-n_{+}\phi^{n_{+}-1})\right]\\!\left[-(\bar{\phi}^{n_{-}}-\bar{\phi}^{n_{+}})+\frac{(\phi+\bar{\phi})}{3\,\alpha}(n_{-}\bar{\phi}^{n_{-}-1}-n_{+}\bar{\phi}^{n_{+}-1})\right]}{3\alpha\,\left(\phi^{{n_{-}}}-\phi^{n_{+}}\right)\left(\bar{\phi}^{n_{-}}-\bar{\phi}^{n_{+}}\right)}\,.$

We compute the eigenvalues of the squared mass terms of the scalar field
$\phi$ using the following expression for the Hessian

$\displaystyle\mathcal{H}=\left[\begin{array}[]{ll}K^{j\bar{l}}\,\nabla_{i}\nabla_{\bar{l}}V&K^{l\overline{m}}\,g_{\overline{m}i}\,\nabla_{l}\nabla_{j}V\\\\[10.0pt]
K^{i\bar{l}}\,g^{j\overline{m}}\,\nabla_{\bar{l}}\nabla_{\overline{m}}\,V&K^{i\bar{l}}\,\nabla_{\bar{l}}\nabla_{j}V\end{array}\right],$
(16)

where
$\nabla_{i}\nabla_{j}\,V=\partial_{i}\,\partial_{j}\,V-\Gamma^{k}_{ij}\,\partial_{k}\,V$
and the Christoffel symbols are

$\Gamma^{\phi}_{\phi\phi}=\Gamma^{\bar{\phi}}_{\bar{\phi}\bar{\phi}}=-\frac{2}{\phi+\bar{\phi}}.$
(17)

In general $V_{i}$ can be simply calculated from

$\displaystyle\partial_{i}\,V=V_{i}=e^{G}\left[G_{i}\,(X-3)+X_{i}\right],$
(18)

for our case it reduces to

$\displaystyle\partial_{i}V$ $\displaystyle=$
$\displaystyle-3\left[a^{2}(-1+3\alpha)\phi^{-2-3\,(\sqrt{\alpha}+\alpha)/2}\bar{\phi}^{-1-3\,(\sqrt{\alpha}+\alpha)/2}(\phi-\bar{\phi})(\phi+\bar{\phi})^{1-3\alpha}\right],$
(19)

which vanishes for

$\displaystyle\phi=\bar{\phi},$ (20)

making evident that the no-scale potential $V$ is flat along the real
direction. The first panel of Fig. 1 shows this scalar potential for
$\alpha=1$, where we compare it to the case when the imaginary direction of
$\phi$ has been stabilized using Eq. (10), second panel of Fig. 1. The
Hessian, Eq. (16) for $\bar{\phi}=\phi$ acquires a simple form because we have
just one pair of fields and hence
$K^{\phi\bar{\phi}}\,\nabla_{\bar{\phi}}\nabla_{\phi}V=K^{\bar{\phi}\phi}\,\nabla_{\phi}\nabla_{\bar{\phi}}V$
. Also the elements above and below the diagonal are equal when
$\bar{\phi}=\phi$, as expected from the form of the potential hence producing
one zero eigenvalue. The non-zero eigenvalue then is just the trace of the
Hessian 222We note that with the substitutions of $\zeta=2\phi$,
$\lambda_{1}=2^{-n_{-}}a$ and $\lambda_{2}=2^{-n_{+}}a$, we recover the result
of Eq. 23 of [42]., corresponding to the squared mass along the imaginary
direction of $\phi$

$\displaystyle m^{2}_{\rm{Im}\
\phi}=\frac{2^{2-3\,\alpha}a^{2}}{\alpha}\,\phi^{-3\sqrt{\alpha}}\left[\,\alpha\left(-1+\
\,\phi^{3\,\sqrt{\alpha}}\right)^{2}-\left(1+\,\phi^{3\,\sqrt{\alpha}}\right)^{2}\right].$
(21)

Figure 1: First row: comparison of the scalar potential, $V(\phi)$ (left) of
Eq. (12), to the potential $V(\phi)$ but with
$K=-3\sum_{i}^{3}\,\alpha_{i}\ln\left(\phi_{i}+\overline{\phi}_{i}+b_{i}(\phi_{i}-\overline{\phi}_{i})^{4}\right)$,
Eq. (10), which stabilizes the imaginary direction. Second row: the potential
$V(x+i\,\sigma)$ for $x$ and the canonical field $\sigma$. The direction $x$
gets fixed to $b$ in order to work with a canonical field, $\sigma$. For all
cases we used $\alpha=1$.

It is well known that for $\alpha\leq 1$ there is an instability for any field
value (e.g. [42]), while for $\alpha>1$ the non-zero eigenvalue is negative
for

$\displaystyle\left[\frac{\sqrt{\alpha}-1}{1+\sqrt{\alpha}}\right]^{\frac{1}{3\sqrt{\alpha}}}<\phi<\left[\frac{1+\sqrt{\alpha}}{\sqrt{\alpha}-1}\right]^{\frac{1}{3\sqrt{\alpha}}}.$
(22)

In order to turn the imaginary component into canonical form, we need to
stabilize the flat real direction. It can be done, for instance, through
modifying the Kähler potential (see [43] for a review) of the model Eq. (2) by

$\displaystyle K$ $\displaystyle=$
$\displaystyle-3\,\alpha\,\ln\left[\phi+\bar{\phi}+\,\frac{\left(\phi+\bar{\phi}-2\,b\right)^{4}}{L}\right],$
(23)

which stabilizes the real component to $\bar{\phi}=\phi=b$, and provides a way
to work only with one canonical field, $\sigma$, proportional to the imaginary
direction (see Appendix A):

$\displaystyle\sigma=\sqrt{\frac{3\,\alpha}{2\,b^{2}}}\,y.$ (24)

The theory of Eq. (2) can be extended to include more fields and stabilize the
real directions with the same procedure. With the addition of Eq. (23) the
scalar potential has the same form of Eq. (12). Note that when fixing the real
part to $b$ all the dependence inversely proportional to $L$ in Eq. (23)
vanishes from the potential and their subsequent derivatives, as can be seen
from Eq. (12), the form of the first derivative, Eq. (19), and of the second
derivatives of $V$,

$\displaystyle\partial_{i}\,\partial_{\bar{\jmath}}\,V=V_{i\,\bar{\jmath}}$
$\displaystyle=$ $\displaystyle
e^{G}\,\left[G_{\bar{\jmath}}\left(G_{i}\,(X-3)+X_{i}\right)+G_{i\,\bar{\jmath}}(X-3)+X_{i\,\bar{\jmath}}+G_{i}\,X_{\bar{\jmath}}\right]\,,$
$\displaystyle\partial_{i}\,\partial_{j}\,V=V_{i\,j}$ $\displaystyle=$
$\displaystyle
e^{G}\,\left[G_{j}\left(G_{i}\,(X-3)+X_{i}\right)+G_{i\,j}(X-3)+X_{ij}+G_{i}\,X_{j}\right].$
(25)

It is clear from these expressions that up to the second derivatives in $\phi$
or $\bar{\phi}$, $G$ (e.g. $G_{ij}$) and X preserve the property of the Kähler
potential in Eq. (23) to reduce to Eq. (2) when $\bar{\phi}=\phi=b$. In the
third panel of Fig. 1 we compare $V$ as a function of $V(x+i\sigma)$ to the
model described by Eq. (83) where the real direction has not been fixed (first
panel) and to the case where the imaginary direction has been stabilized (left
panel). We have chosen to plot $V(x+i\sigma)$ as we can clearly see the
behaviour when $\phi=\bar{\phi}=b$. In Fig. 2 we plot the potential $V$ as a
function of $\sigma$ for values of $\alpha$ which are perfect squares modulo
9, where we can see the transitions from maxima to minima: values for
$\alpha\leq 1$ give de Sitter maxima, while values for $\alpha>1$ de Sitter
minima.

For the case of having $\alpha=\frac{n^{2}}{9}$, $n$ an integer, using Eq.
(21), then the second criterion of the RdSC, Eq. (2), can be written as
$-m^{2}_{\rm Im[\phi]}/V-c^{\prime}\geq 0$ or equivalently
$\left[\frac{9}{n^{2}}-1\right]b^{2n}+\left[\frac{18}{n^{2}}+2-3c^{\prime}\right]\,b^{n}+\frac{9}{n^{2}}-1\geq
0$. For $n>3$ we find that this bound can be satisfied for $b$ such that
$a_{-}<b^{n}<a_{+}$ with
$a_{\pm}=\frac{9+n^{2}-3c^{\prime}n^{2}/2\,\pm\,n^{2}\,\sqrt{\left(2-3c^{\prime}/2\right)\left(18/n^{2}-3c^{\prime}/2\right)}}{n^{2}-9}$
when $c^{\prime}\leq\frac{12}{n^{2}}<\frac{4}{3}$. For $n<3$ instead it can be
satisfied for all values of $b$ if $\frac{4}{3}\leq
c^{\prime}\leq\frac{12}{n^{2}}$. Finally for the special case $n=3$, the bound
is satisfied for all values of $b$ if $c^{\prime}\leq\frac{4}{3}$.

Figure 2: The scalar potential, $V(\sigma)$, of Eq. (12) with the Kähler
potential of Eq. (23) with the stabilization for
$\langle{\rm{Re}}\,[\phi]\rangle=b=1$, as a function of the canonical field
$\sigma=\sqrt{3\,\alpha/2}\ y/b$. We have chosen values of $\alpha$ which are
perfect squares modulo 9. Values for $\alpha\leq 1$, left, give de Sitter
maxima, while values for $\alpha>1$, right, can admit de Sitter minima.

### Trans-Planckian Conjecture in No-Scale de Sitter Models

The TCC implies that a de Sitter phase has a limited lifetime. In the case of
a potential with de Sitter maximum, the field can sit on that critical point
if the negative curvature, second derivative of the scalar potential, is
bounded from below. Specifically, close to a local maximum if there is field
range $0<\sigma<\Delta\sigma$ where
$|V^{\prime\prime}|\leq|V^{\prime\prime}|_{\rm max}$, defining
$V^{\prime\prime}\equiv\partial^{2}_{\sigma}V$, the TCC is satisfied if either
of the following conditions are satisfied [32]

$\displaystyle\Delta\sigma$ $\displaystyle<$
$\displaystyle\frac{\frac{2}{3\sqrt{\pi}}\left(V_{0}\,V_{\rm{min}}\right)^{3/4}\,\ln^{1/2}\sqrt{\frac{3}{V_{\rm{min}}}}}{\frac{2}{3}V(\sigma)-|V^{\prime\prime}|_{\rm{max}}\,\ln^{2}\sqrt{\frac{3}{V_{\rm{min}}}}},$
(26) $\displaystyle\frac{|V^{\prime\prime}|_{\rm{max}}}{V_{\rm{min}}}$
$\displaystyle\geq$
$\displaystyle\frac{2}{3}\frac{1}{\ln^{2}\sqrt{\frac{3}{V_{\rm{min}}}}}.$ (27)

In the above equations $|V^{\prime\prime}|_{\max}$, is the maximum curvature
and $V_{\rm{min}}$ is the minimum value for the potential in that field range.
The second inequality is similar to the second criterion, RdSC (2), with a
prediction for $c^{\prime}$ up to a logarithmic correction and thus it is a
milder constraint. For the dS no-scale model we are considering and for the
single field range, the inequality Eq. (26) implies that

$\displaystyle\frac{2}{3\sqrt{\pi}}\left(V_{0}V_{\rm
min}\right)^{3/4}\ln^{1/2}\sqrt{\frac{3}{V_{\rm{min}}}}-\left[\frac{2}{3}V_{\rm
min}-|V^{\prime\prime}|_{\rm{max}}\ln^{2}\sqrt{\frac{3}{V_{\rm{min}}}}\right]\sigma>0.$
(28)

For $\alpha\leq 1$ the expression in the square bracket is negative and the
bound is satisfied for any value of $\sigma$. In order to see that, we note
$V_{\rm min}\leq V_{0}$ and $|V^{\prime\prime}|_{\rm
max}\geq|V^{\prime\prime}_{0}|$, therefore

$\displaystyle\frac{2}{3}V_{\rm{min}}-|V^{\prime\prime}|_{\rm
max}\ln^{2}\sqrt{\frac{3}{V_{\rm{min}}}}\\!\\!$ $\displaystyle<$
$\displaystyle\\!\\!\frac{2}{3}V_{0}-|V^{\prime\prime}_{0}|\,\ln^{2}\sqrt{\frac{3}{V_{0}}}$
$\displaystyle=$ $\displaystyle
2^{3-3\alpha}a^{2}\left[1-\frac{b^{-3\sqrt{\alpha}}}{2\alpha}\left((1+b^{3\sqrt{\alpha}})^{2}-\alpha(1-b^{3\sqrt{\alpha}})^{2}\right)\ln^{2}(2^{1-\frac{3\alpha}{2}}a)\right]\,,$

where in the second line we have used Eq. (21) which can be written in the
form

$\displaystyle
V^{\prime\prime}|_{\sigma=0}=-2^{2-3\alpha}a^{2}\frac{b^{-3\sqrt{\alpha}}}{\alpha}\left[(1+b^{3\sqrt{\alpha}})^{2}-\alpha(1-b^{3\sqrt{\alpha}})^{2}\right].$
(30)

For $\alpha<1$ the above expression is negative for values of $b$ that make
the potential unstable. For $\alpha=1$, the above expression is negative for
$a<\sqrt{2}e^{\frac{-1}{\sqrt{2}}}$ or $a>\sqrt{2}e^{\frac{1}{\sqrt{2}}}$
independently of $b$, as can be seen from the left panel of Fig. 3. We recall
that for all phenomenological purposes $a\ll 1$. For $\alpha>1$, the
analytical computation is involved as can be seen from the right panel of Fig.
3. Numerical inspection then give us for this case the values of $\sigma$ and
$b$ that are compatible with the TCC bound, Eq. (28).

Figure 3:  Evaluation of the TCC for $\alpha=1$, left plot, and $\alpha=25/9$,
right plot. The blue regions indicate values of $b$ and $\sigma$ which are
compatible with the TCC bound when $V>0$. The orange region indicates values
for which $V<0$. For both plots we have used $a=0.1$. Note that for the case
of $\alpha=1$ the value of $b$ can extend for all values along the real line.

The TCC conjecture also allows the existence of de Sitter potentials with
metastable minima as long as these minima are not positive or if positive by
allowing quantum tunnelling. This allows sufficiently short-lived transient
quasi-dS like phases. We note that potentials with negative minima could be
constructed from the kind of models of the right-side panel of Fig. 3, with
addition of other fields that render the minima negative.

##  3 No-Scale Models with Rolling Dynamics

In this section, we first add a rolling field to our minimal model, then we
obtain the conditions for finding extrema, point out in which cases there can
be a stable minimum, without the addition of a quartic term in the Kähler
potential. Then we point out in which cases instead we can be in agreement
with the first criterion of the RdSC, Eq. (1).

Following the approach proposed in [44], we modify the Kähler potential and
the superpotential of our basic dS no-scale model as follows

$\displaystyle\widehat{K}$ $\displaystyle=$ $\displaystyle
K-\sum_{m=1}^{N}\,q_{m}\ln(\chi_{m}+\bar{\chi}_{m}),$ (31)
$\displaystyle\widehat{W}$ $\displaystyle=$ $\displaystyle
W\,\prod_{m=1}^{M}\,\left(c_{0,m}+c_{1,m}\,\chi_{m}^{-{\nicefrac{{p_{m,1}}}{{2}}}}+c^{2}_{m}\,\chi_{m}^{-{\nicefrac{{p_{m,2}}}{{2}}}}+\cdots\right).$
(32)

Here $m$ runs over the number of rolling chiral superfields, $q_{m}>0$,
$c_{k,m}$ are arbitrary constants and the powers $p_{m,k}$ satisfy
$p_{m,k}<0$. The function invariant under Kähler transformations
$\widehat{G}=\widehat{K}+\ln\ \left|\widehat{W}\right|^{2}$ of the modified
model is given by

$\widehat{G}=G+\sum_{m=1}^{M}\,\left(\ln\frac{|c^{0}_{m}+c^{1}_{m}\,\chi_{m}^{-{\nicefrac{{p_{m}^{1}}}{{2}}}}+c^{2}_{m}\,\chi_{m}^{-{\nicefrac{{p_{m}^{2}}}{{2}}}}+\cdots|^{2}}{(\chi_{m}+\bar{\chi}_{m})^{q_{m}}}\right)\equiv
G+\widetilde{G},$ (33)

where additional chiral superfields $\chi_{m}$ are included to provide an
escape to the RdSC for positive scalar potentials, Eq. (4). The addition of
these chiral superfields solves at the same time the instability of $K$ at Im
$\phi$=0, for which V has a maximum, Eq. (5) without additional quartic terms.
The scalar potential for this model can be written then as follows

$\displaystyle\widehat{V}$ $\displaystyle=$ $\displaystyle
e^{\widehat{G}}\left(\sum_{I=1,\bar{J}=1}^{N+M,N+M}\widehat{G}_{I}\,\widehat{K}^{I\bar{J}}\,\widehat{G}_{\bar{J}}-3\right)$
(34) $\displaystyle=$ $\displaystyle
e^{\widehat{G}}\left(\sum_{i=1,\bar{\jmath}=1}^{N,N}\widehat{G}_{i}\,\widehat{K}^{i\bar{\jmath}}\,\widehat{G}_{\bar{\jmath}}-3\right)+e^{\widehat{G}}\left(\sum_{m=1,\bar{n}=1}^{M,M}\widehat{G}_{m}\,\widehat{K}^{m\bar{n}}\,\widehat{G}_{\bar{n}}\right)$
$\displaystyle=$ $\displaystyle
e^{\widetilde{G}}\,\left(V+\widehat{X}\,{\rm{e}}^{G}\right),$

where $I=i,m$ runs over all chiral superfields, indices $i,j=1,\ldots,N$ are
for the de Sitter fields, indices $m,n=1,\ldots,M$ are devoted to the rolling
fields, $\widehat{K}^{I\bar{J}}$ is the inverse of the Kähler metric (which is
block diagonal)
$\widehat{K}_{I\bar{J}}=\partial_{I}\partial_{\bar{J}}\,\widehat{K}$ and
$\widehat{G}_{I}=\partial_{I}\,\widehat{G}$, analogously for its Hermitian
conjugate. The parameter $\widehat{X}$ is defined as

$\widehat{X}\equiv\widehat{G}_{m}\,\widehat{K}^{m\bar{n}}\,\widehat{G}_{\bar{n}}=\widetilde{G}_{m}\,\widehat{K}^{m\bar{n}}\,\widetilde{G}_{\bar{n}},$
(35)

where the expression after the equal sign follows from the decomposition in
Eq. (33) and $\widehat{G}_{m}=\widetilde{G}_{m}$ since only $\widetilde{G}$
depends on the “rolling” fields $\chi$. For the rest of this section we assume
just one rolling field $\chi$. The potential $\widehat{V}$ is minimal in the
$\chi$ plane for $\rm{Im}\,[\chi]=0$, therefore we consider only the real part
of the scalar field

$\displaystyle\gamma=\left.\widehat{X}\right|_{\widehat{V}_{\rm{Re}\,[\chi]}}.$
(36)

The extremization condition of the new scalar potential, $\widehat{V}$, is

$\displaystyle\partial_{i}\,\widehat{V}=e^{\widetilde{G}}\,\left(\partial_{i}\,V+\gamma\
\partial_{i}\,e^{G}\right)=e^{\widetilde{G}}\,\left(\partial_{i}\,V+\gamma\
\partial_{i}\,(e^{K}|W|^{2})\right)=0.$ (37)

This equation can be satisfied in two ways. One way is requiring both terms to
cancel independently, since $\partial_{\bar{i}}V=0$ and
$\partial_{\bar{\imath}}\,V=0$ are the conditions for the minimum of the
original theory this can be easily satisfied and hence the term
$\partial_{i}\,(e^{K}|W|^{2})$ should be satisfied independently. The other
way is to achieve a cancellation with both terms.

In the case of finding solutions where $W=0$, the solutions for
$\partial_{i}\,V$, Eq. (18), can be written as follows

$\displaystyle\partial_{i}\,V=e^{K}\,\overline{W}_{\bar{\jmath}}\left[K^{\bar{\jmath}l}\left(W_{l}\,K_{i}+W_{i}\,K_{l}+W_{li}\right)+K^{\bar{\jmath}l}\,_{i}W_{l}\right]=0.$
(38)

When the addition of the rolling field does not change the minima of the no-
scale field in the supersymmetry breaking vacuum, then the no-scale field is
stabilized and the only dynamics comes from the rolling field, as it has been
emphasized in [44]. For the second term in Eq. (37) we have

$\displaystyle\partial_{i}\left(e^{K}|W|^{2}\right)=e^{K}\
\overline{W}\left(WK_{i}+W_{i}\right),$ (39)

for which we have two solutions for a vanishing term

$\displaystyle W=0\quad\text{or}\quad D_{i}\,W=K_{i}+\frac{W_{i}}{W}=0,$ (40)

analogously for their hermitian counterparts. The first solution in Eq. (40)
can be satisfied in general, while the second solution just holds for certain
values of $\alpha$. We choose the first one, so we have at the extrema

$\displaystyle\prod_{i=1}^{N}\,\phi_{i}^{n_{i+}}$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{N}\,\phi_{i}^{n_{i-}},$
$\displaystyle\prod_{i=1}^{N}\,\overline{\phi}_{i}^{n_{i+}}$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{N}\,,\overline{\phi}_{i}^{n_{i-}},$ (41)

where non trivial solutions (for which $n_{i+}\neq n_{i-}$) exist, as we will
see in the next sections, hence

$\displaystyle\left.\partial_{\bar{\imath}}\left(e^{K}|W|^{2}\right)\right|_{\prod_{i=1}^{N}\,\overline{\phi}_{i}^{n_{i}+}=\prod_{i=1}^{N}\,\overline{\phi}_{i}^{n_{i}-}}=0,$
(42)
$\displaystyle\left.\partial_{i}\left(e^{K}|W|^{2}\right)\right|_{\prod_{i=1}^{N}\,\phi_{i}^{n_{i}+}=\prod_{i=1}^{N}\,\phi_{i}^{n_{i}-}}=0.$

Finally, we note that the evaluation of $\widehat{V}$ at the extrema for the
no-scale fields implies

$\left.\
\frac{K^{I\bar{J}}\,\partial_{I}\widehat{V}\,\partial_{\bar{J}}\,\widehat{V}}{\widehat{V}^{2}}=\
\frac{\widehat{K}^{m\bar{n}}\partial_{m}\,\widehat{V}\partial_{\bar{n}}\,\widehat{V}}{\widehat{V}^{2}}\right|_{\rm{\widehat{V}_{\rm{ext.}}}}=\gamma\,,$
(43)

where we have used $\partial_{i}\,\widehat{V}=0$. We observe that the rolling
behavior in this no-scale supergravity puts the model out of the swamp given
that $2\gamma\geq c^{2}$. It is interesting to note, as the superpotential is
vanishing in the vacuum, that we get a supersymmetry breaking vacuum with
positive vacuum energy for any value of $\gamma$. This can be seen by
expanding the scalar potential $\widehat{V}$ into pieces of the original model
and the pieces coming from the rolling field

$\displaystyle\widehat{V}={e}^{\widetilde{G}}\,{e}^{K}|W|^{2}\,\left(X-3+\gamma\right)={e}^{\widetilde{G}}\,e^{K}\,\left(\sum_{i=1}^{N}\,\left|D_{i}\,W\right|^{2}+(\gamma-3)|W|^{2}\right),$
(44)

from which it follows that at the extrema of the complete model the dependence
on $\gamma$ disappears when $W=0$

$\displaystyle\widehat{V}=\left.e^{\widetilde{G}}\,\left(\,e^{K}\,\sum_{i=1}^{N}\,|D_{i}\,W|^{2}\,\right)\right|_{\rm
ext.}.$ (45)

### The Minimal Model

In this section we study the minimal model including a no-scale chiral
superfield plus a rolling superfield.

#### Existence of Extrema

We start with the simplest example including one no-scale chiral superfield
and one rolling chiral superfield. The Kähler and the superpotential are as
follows

$\displaystyle\widehat{K}$ $\displaystyle=$
$\displaystyle-3\,\alpha\,\ln(\phi+\bar{\phi})-q\,\ln(\chi+\bar{\chi}),$
$\displaystyle\widehat{W}$ $\displaystyle=$ $\displaystyle
a\left(\phi^{n_{+}}-\phi^{n_{-}}\right)\,\chi^{-p/2},$ (46)

where $\alpha,q>0,p<0$, $a$ is an arbitrary constant and
$n_{\pm}=3/2(\alpha\pm\sqrt{\alpha})$. The full scalar potential,
$\widehat{V}$, is as in Eq. (34), $V$, the no-scale potential is given in Eq.
(12) and $\widetilde{G}$ is

$\displaystyle{\widetilde{G}}$ $\displaystyle=$ $\displaystyle\ln\left[({2\rm
Re}\,[\chi])^{-q}\,|\chi|^{-p}\right],$ (47)

For the second term Eq. (37) the extremization conditions of Eq. (42), for the
non-trivial solution (that is $n_{i-}\neq n_{i+}$), and for an arbitrary value
of $\alpha$ reduce to

$\left.\phi\right|_{\rm ext.}=\left.\bar{\phi}\right|_{\rm ext.}=1,$ (48)

which is in agreement with the condition for finding the extrema of the
original theory, Eq. (20), but restricts the set of possible values for the
extrema. If we instead consider the Kähler potential of Eq. (23), this can
automatically fix $\phi=\bar{\phi}=1$ by choosing $b=1$ as explained in
section 2.

#### Stability Condition

We study the stability of the vacuum by considering the Hessian matrix

$\displaystyle\mathcal{H}=\left[\begin{array}[]{ll}K^{J\bar{L}}\,\nabla_{I}\nabla_{\bar{L}}\widehat{V}&K^{L\overline{M}}\,g_{\overline{M}I}\,\nabla_{L}\nabla_{J}\widehat{V}\\\\[10.0pt]
K^{I\bar{L}}\,g^{J\overline{M}}\,\nabla_{\bar{L}}\nabla_{\overline{M}}\,\widehat{V}&K^{I\bar{L}}\,\nabla_{\bar{L}}\nabla_{J}\widehat{V}\,,\end{array}\right]\,,$
(51)

where $I=\phi,\chi$ and
$\widehat{V}_{IJ}=\nabla_{I}\,\nabla_{J}\,\widehat{V}=\partial_{I}\,\partial_{J}\widehat{V}-\Gamma^{K}_{IJ}\,\partial_{K}\,\widehat{V}$.
For the no-scale extremization where $\partial_{\phi}V=0$ and
$e^{G}\sim|W|^{2}=0$ we find that

$\displaystyle\left.\widehat{V}_{\phi\chi}\,\right|_{\rm
min}=\left(\partial_{\chi}\,e^{\widetilde{G}}\right)\,\left(\partial_{\phi}V\right)+\gamma\,e^{\widetilde{G}}\,e^{G}(G_{\phi}\,G_{\chi})=0,$
(52)

similarly for any other mixed component and its Hermitian conjugate. The no-
scale components do not mix with the rolling field, thus the mixed Christoffel
symbols are vanishing. Therefore, we can write the Hessian matrix as

$\displaystyle\left.{\mathcal{H}}\right|_{\phi=\bar{\phi}=1}=\left.\begin{bmatrix}\widehat{K}^{\phi\bar{\phi}}\,\widehat{V}_{\phi\bar{\phi}}&0&\widehat{K}^{\phi\bar{\phi}}\,g_{\bar{\phi}\phi}\,\widehat{V}_{\phi\phi}&0\\\
0&\widehat{K}^{\chi\bar{\chi}}\,\widehat{V}_{\chi\bar{\chi}}&0&\widehat{K}^{\chi\bar{\chi}}\,g_{\bar{\chi}\chi}\,\widehat{V}_{\chi\chi}\\\
\widehat{K}^{\phi\bar{\phi}}\,g^{\bar{\phi}\phi}\,\widehat{V}_{\bar{\phi}\bar{\phi}}&0&\widehat{K}^{\phi\bar{\phi}}\widehat{V}_{\bar{\phi}\phi}&0\\\
0&\widehat{K}^{\chi\bar{\chi}}\,g^{\bar{\chi}\chi}\,\widehat{V}_{\bar{\chi}\bar{\chi}}&0&\widehat{K}^{\chi\bar{\chi}}\,\widehat{V}_{\bar{\chi}\chi}\end{bmatrix}\right|_{\phi=\bar{\phi}=1},$
(53)

from which we can see that it is equivalent to look for the eigenvalues, which
need to be positive for the stability of the scalar potential, of the
following block-diagonal matrix

$\displaystyle{\mathcal{H}}=\left[\begin{array}[]{cc}\mathcal{H}_{\phi}&0\\\
0&\mathcal{H}_{\chi}\end{array}\right].$ (56)

Each of the blocks in Eq. (56) contain only information from one field:

$\displaystyle\rm{det}\left[\mathcal{H}_{\phi}\right]=\rm{det}\left[K^{\phi\bar{\phi}}\,\begin{bmatrix}\widehat{V}_{\phi\bar{\phi}}&\widehat{V}_{\phi\phi}\\\
\widehat{V}_{\bar{\phi}\bar{\phi}}&\widehat{V}_{\bar{\phi}\phi}\end{bmatrix}\right],\quad\rm{det}\left[\mathcal{H}_{\chi}\right]=\rm{det}\left[K^{\chi\bar{\chi}}\,\begin{bmatrix}\widehat{V}_{\chi\bar{\chi}}&\widehat{V}_{\chi\chi}\\\
\widehat{V}_{\bar{\chi}\bar{\chi}}&\widehat{V}_{\bar{\chi}\chi}\end{bmatrix}\right],$
(57)

also, due to having only one field in each sector, and
$\widehat{K}^{\phi\bar{\phi}}=K^{\phi\bar{\phi}}$,
$\widehat{K}^{\chi\bar{\chi}}=K^{\chi\bar{\chi}}$, the factor of the Kähler
metric factors out. We therefore can look independently for the eigenvalues of
the sub-matrices $\mathcal{H}_{\phi}$ and $\mathcal{H}_{\chi}$. The second
derivatives of the scalar potential, Eq. (34), at the local extrema are given
as follows

$\displaystyle\widehat{V}_{\phi\phi}$ $\displaystyle=$ $\displaystyle
e^{\widetilde{G}}\,\left(\partial_{\phi}\,\partial_{\phi}V\right)+\gamma\,e^{\widehat{G}}\,\left(K_{\phi\phi}+G_{\phi}G_{\phi}\right)-\Gamma^{\phi}_{\phi\phi}\,\partial_{\phi}\widehat{V}=e^{\widetilde{G}}\left(2^{-3\alpha}\,\times
6\,a^{2}\right),$ $\displaystyle\widehat{V}_{\phi\bar{\phi}}$ $\displaystyle=$
$\displaystyle
e^{\widetilde{G}}\,\left(\partial_{\phi}\,\partial_{\bar{\phi}}V\right)+\gamma\,e^{\widehat{G}}\,(K_{\phi\bar{\phi}}+G_{\phi}G_{\bar{\phi}})=e^{\widetilde{G}}\left(-2^{-3\alpha}\,\times
6\,a^{2}+2^{-3\alpha}\,\times 9\,\gamma\,\alpha\,a^{2}\right),$
$\displaystyle\widehat{V}_{\bar{\phi}\bar{\phi}}$ $\displaystyle=$
$\displaystyle
e^{\widetilde{G}}\,\left(\partial_{\bar{\phi}}\,\partial_{\bar{\phi}}V\right)+\gamma\,e^{\widehat{G}}\,(K_{\bar{\phi}\bar{\phi}}+G_{\bar{\phi}}G_{\bar{\phi}})-\Gamma^{\bar{\phi}}_{\bar{\phi}\bar{\phi}}\,\partial_{\bar{\phi}}\widehat{V}=e^{\widetilde{G}}\left(2^{-3\alpha}\,\times
6\,a^{2}\right),$ (58)

where $e^{\widetilde{G}}$ is given in Eq. (47) and the Christoffel symbols in
Eq. (17). We note that $e^{\widehat{G}}\,G_{\phi}\,G_{\bar{\phi}}\neq 0$.
Finally, the eigenvalues of the Hessian $\mathcal{H}_{\phi}$ at
$\phi=\bar{\phi}$ are

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\text{e}^{\widetilde{G}}\,\frac{4}{3\,\alpha}\,\left[9\,\times
2^{-3\,\alpha}\,a^{2}\,\alpha\,\gamma\right]=3\times\text{e}^{\widetilde{G}}\times
2^{2-3\,\alpha}\,a^{2}\,\gamma,$ $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\text{e}^{\widetilde{G}}\,\frac{4}{3\,\alpha}\,\left[3\,\times
2^{-3\,\alpha}\,a^{2}(-4+3\,\alpha\gamma)\right]=\text{e}^{\widetilde{G}}\times
2^{2-3\,\alpha}\,a^{2}\left(-\frac{4}{\alpha}+3\,\gamma\right).$ (59)

Therefore, the stability condition for having a de Sitter minimum implies

$\displaystyle 3\,\alpha\,\gamma>4.$ (60)

This shows that adding a rolling dynamics, without adding a quartic term in
the Kähler metric to stabilize the imaginary part, Eq. (10), also provides a
way to stabilize the potential. In case presented in this section, the
eigenvalues of $\lambda_{1}$ and $\lambda_{2}$, Eq. (3) can be both positive
for suitable values of $\alpha$ and $\gamma$.

#### Assertion of the Refined de Sitter Conjecture

Now we check whether the modified no-scale model obeys the first criterion of
RdSC, Eq. (1), or not. We first compute the parameter $\gamma$, from Eq. (35)

$\gamma=\left.G_{\chi}\,K^{\chi\bar{\chi}}\,G_{\bar{\chi}}\right|_{{\rm
Im}\chi=0}=\left.\frac{1}{q|\chi|^{2}}\left[q^{2}|\chi|^{2}+\left(\frac{p}{2}\right)^{2}\
|\chi+\overline{\chi}|^{2}+q\
p\left(\chi+\overline{\chi}\right)^{2}\right]\right|_{{\rm
Im}\chi=0}=\frac{(p+q)^{2}}{q},$ (61)

where

$G_{\chi}=-\frac{p}{2\chi}-\frac{q}{\chi+\bar{\chi}},\quad{\rm and}\quad
G_{\bar{\chi}}=-\frac{p}{2\bar{\chi}}-\frac{q}{\chi+\bar{\chi}},$ (62)

and $K^{\chi\bar{\chi}}$ is the inverse of the Kähler metric

$K_{\chi\bar{\chi}}=\frac{q}{(\chi+\bar{\chi})^{2}}.$ (63)

The first criterion of RdSC, Eq. (1), in the form of Eq. (4), is satisfied if

$\frac{\sqrt{2}(p+q)}{\sqrt{q}}\geq c={\cal O}(1).$ (64)

Along the real direction, we have

$e^{\widetilde{G}}=|\chi\,\bar{\chi}|^{-p/2}(\chi+\bar{\chi})^{-q}=({\rm
Re}\,\chi)^{-(p+q)}=e^{\sqrt{2\gamma}\,\chi^{c}},$ (65)

where $\chi^{c}$ is the canonically normalized rolling scalar field. Finally,
we note that if the first criterion is violated, the second criterion of the
RdSC, Eq. (2), can be satisfied if there is an instability. In fact, the
second mass eigenvalue is negative for $3\,\alpha\,\gamma<4$. Given that
$\widehat{V}=12\,a^{2}\,\times 2^{-3\,\alpha}e^{\sqrt{2\,\gamma}\,\chi^{c}}$
in the real field direction, the second criterion of the RdSC, Eq. (2),
requires that

$\displaystyle 3\,\alpha\,\gamma\leq 4-\alpha\,c^{\prime}.$ (66)

Since we computed the Hessian for non-canonical fields, but kept the Kähler
metric in Eq. (51) we can evaluate the second criterion, Eq. (2), involving
the fields $\phi$ and $\chi$, without the need of using canonical fields.

#### Superpotential with two Terms in the Expansion of the Rolling Field

Now we modify the superpotential Eq. (3) of the minimal model to take the
following form

$W=(\phi^{n_{+}}-\phi^{n_{-}})(a_{1}\,\chi^{-p_{1}/2}+a_{2}\,\chi^{-p_{2}/2}).$
(67)

From Eq. (33) we compute

$e^{\widetilde{G}}=(\chi+\bar{\chi})^{-q}\,\left|a_{1}\chi^{-p_{1}/2}+a_{2}\,\chi^{-p_{2}/2}\right|^{2},$
(68)

where

$G_{\chi}=-\frac{p_{1}+q}{2\,\chi}-\frac{(p_{2}-p_{1})\,a_{2}}{2\,\chi(a_{1}\,\chi^{(p_{2}-p_{1})/2}+a_{2})},$
(69)

to finally get

$\gamma=\frac{(p_{1}+q)^{2}}{q}+\frac{a_{2}\,(p_{2}-p_{1})}{q}\left[\frac{a_{2}\,(p_{2}-p_{1})}{a_{1}\,\chi^{(p_{2}-p_{1})/2}+a_{2}}+2(p_{1}+q)\right]\frac{1}{a_{1}\,\chi^{(p_{2}-p_{1})/2}+a_{2}}.$
(70)

From Eq. (70) we observe that as far as $\sqrt{2\gamma}\geq c$ the RdSC, in
the form of Eq. (4), can be satisfied.

##  4 Non-minimal No-Scale Models

##### 2+1 Model

Next, we consider a no-scale supergravity with dS vacuum generated by two
chiral superfields $\phi_{i}$ and add one rolling superfield $\chi$, we refer
to this construction as the _2+1 Model_. The Kähler and the superpotential are
as follows

$\displaystyle K$ $\displaystyle=$
$\displaystyle-3\,\alpha_{1}\ln(\phi_{1}+\bar{\phi}_{1})-3\,\alpha_{2}\,\ln(\phi_{2}+\bar{\phi}_{2})-q\,\ln(\chi+\bar{\chi}),$
(71) $\displaystyle W$ $\displaystyle=$ $\displaystyle
a\left(\phi_{1}^{n_{1+}}\,\phi_{2}^{n_{2+}}-\phi_{1}^{n_{1-}}\,\phi_{2}^{n_{2-}}\right)\,\chi^{-p/2},$
(72)

where $\alpha_{i},q>0,p<0$, $a$ is an arbitrary constant and $n_{i\pm}$ are
given in Eq. (7). From Eq. (3) and demanding $n_{i+}\neq n_{i-}$, we have

$\phi_{1}^{r_{1}}\phi_{2}^{r_{2}}=\bar{\phi}_{1}^{r_{1}}\bar{\phi}_{2}^{r_{2}}=1,$
(73)

where $r_{1},r_{2}$ are arbitrary constants satisfying
$r_{1}^{2}+r_{2}^{2}=1$. We note that we cannot have both $\partial_{1}\,W$
and $\partial_{2}\,W$ simultaneously vanishing as supersymmetry is
spontaneously broken and the scalar potential is lifted. However, one of them
can vanish for special case of $r_{1}=0$ and $r_{2}=1$, or vice versa.
Moreover, we observe that the above solution Eq. (73) satisfies Eq. (38).
Similar to the minimal model, the Hessian matrix can be brought to a block
diagonal form, Eq. (56), where now

$\displaystyle{\mathcal{H}}_{\phi}=\left[\begin{array}[]{cccc}K^{1\bar{1}}\,\widehat{V}_{1\bar{1}}&K^{1\bar{1}}\,g_{1\bar{1}}\,\widehat{V}_{11}&K^{1\bar{1}}\,\widehat{V}_{1\bar{2}}&K^{1\bar{1}}\,g_{1\bar{1}}\widehat{V}_{12}\\\
K^{1\bar{1}}\,g^{1\bar{1}}\widehat{V}_{\bar{1}\bar{1}}&K^{1\bar{1}}\,\widehat{V}_{\bar{1}1}&K^{1\bar{1}}\,g^{1\bar{1}}\,\widehat{V}_{\bar{1}\bar{2}}&K^{1\bar{1}}\,\widehat{V}_{\bar{1}2}\\\
K^{2\bar{2}}\,g^{2\bar{2}}\,\widehat{V}_{2\bar{1}}&K^{2\bar{2}}\,g_{2\bar{2}}\,\widehat{V}_{21}&K^{2\bar{2}}\widehat{V}_{2\bar{2}}&K^{2\bar{2}}\,g_{2\bar{2}}\,\widehat{V}_{22}\\\
K^{2\bar{2}}\,g^{2\bar{2}}\,\widehat{V}_{\bar{2}\bar{1}}&K^{2\bar{2}}\,\widehat{V}_{\bar{2}1}&K^{2\bar{2}}\,\,g^{2\bar{2}}\widehat{V}_{\bar{2}\bar{2}}&K^{2\bar{2}}\,\widehat{V}_{\bar{2}2}\end{array}\right],$
(78)

and ${\mathcal{H}}_{\chi}$ is as in Eq. (57), therefore the problem reduces to
analyzing the $4\times 4$ ${\mathcal{H}}_{1}$ matrix of Eq. (78). Analogously
to Eq. (3) we need to compute the second order derivatives of $\widehat{V}$
with respect to $\phi_{1}$ and $\phi_{2}$:
$\widehat{V}_{\phi_{i}\phi_{j}}=\partial_{i}\,\partial_{j}\,{\rm{e}}^{\widetilde{G}}\left(V+\gamma\,e^{G}\right)=$
${\rm{e}}^{\widetilde{G}}\,\partial_{i}\,\partial_{j}\,\left(V+\gamma\,e^{G}\right)$.
These derivatives are given in the Appendix B. We find that the eigenvalues
are given as follows

$\displaystyle\lambda_{1}=0,$ $\displaystyle{e^{-\tilde{G}}}$
$\displaystyle\lambda_{2}=\frac{12\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}(3\,r_{2}^{2}\,\alpha_{1}+3\,r_{1}^{2}\,\alpha_{2}+s^{2}\,\alpha_{1}\,\alpha_{2}(\gamma-3))(r_{2}^{2}\alpha_{1}+r_{1}^{2}\alpha_{2})}{s^{4}\,\alpha_{1}^{2}\,\alpha_{2}^{2}},$
(79)

where $s$ has been defined in Eq. (7) , which requires

$\displaystyle
3\,r_{2}^{2}\,\alpha_{1}+3\,r_{1}^{2}\,\alpha_{2}+s^{2}\,\alpha_{1}\,\alpha_{2}(\gamma-3)>0,$
(80)

while the other two eigenvalues are of the form

$\displaystyle{e^{-\tilde{G}}}\,\lambda_{\pm}=A\pm\sqrt{B},$ (81)

where $A$ and $B$ are independent of $\phi_{1}$ and $\phi_{2}$. In Fig. 4 we
plot the possible values of $\alpha_{1}$ and $\alpha_{2}$ for the special case
of $\gamma=2$ and $c^{\prime}=1$ with $r_{1}=r_{2}=\frac{1}{\sqrt{2}}$ that
are compatible either with the RdSC or the stability condition.

Figure 4: Parameter space of $\alpha_{1}$ and $\alpha_{2}$, for the $2+1$
model for the special case of $\gamma=2$ and $c^{\prime}=1$ with
$r_{1}=r_{2}=\frac{1}{\sqrt{2}}$, that can be either compatible with the RdSC
or the stability condition.

##### N+1 Model

We generalize to the case of N de Sitter case plus a rolling field with the
following general Kähler potential and superpotential

$\displaystyle K$ $\displaystyle=$
$\displaystyle-3\sum_{i=1}^{N}\alpha_{i}\ln(\phi_{i}+\bar{\phi}_{i})-q\ln(\chi+\bar{\chi})=K-q\ln(\chi+\bar{\chi}),$
(82) $\displaystyle W$ $\displaystyle=$ $\displaystyle
a\,\left(\prod_{i=1}^{N}\phi_{i}^{n_{i+}}-\prod_{i=1}^{N}\phi_{i}^{n_{i-}}\right)\,\chi^{-p/2}=W\chi^{-p/2}.$
(83)

In this case, local supersymmetry breaking vacua exist when for some fields
Eq. (3) is satisfied (leading to $W=\overline{W}=0$), which can be put in the
form

$\prod_{i=1}^{N}\phi_{i}^{r_{i}}=\prod_{i=1}^{N}\bar{\phi}_{i}^{r_{i}}=1,$
(84)

where $r_{1},r_{2}$ are arbitrary constants satisfying
$\sum_{i=1}^{N}r_{i}^{2}=1$. We note that we cannot have $\partial_{i}\,W$
simultaneously vanishing for all $i$. However, some of them can vanish for
special values of $r_{i}$. As with the previous examples this solution
automatically satisfies $\partial_{i}\,V=0$, Eq. (38), which requires
$\phi=\bar{\phi}$. The part of the Hessian regarding only $\phi_{i}$ fields
does not mix with that of $\chi_{i}$, hence as in the previous case the total
Hessian is block diagonal, Eq. (56), where

$\displaystyle{\mathcal{H}}_{\phi}=\left[\begin{array}[]{ccccc}\widehat{V}_{\phi_{1}\bar{\phi}_{1}}&\widehat{V}_{\phi_{1}\bar{\phi}_{2}}&\ldots&\ldots&\widehat{V}_{\phi_{1}\phi_{N}}\\\
\widehat{V}_{\phi_{2}\bar{\phi}_{1}}&\widehat{V}_{\phi_{2}\bar{\phi}_{2}}&\ldots&\ldots&\widehat{V}_{\phi_{2}\phi_{N}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
\widehat{V}_{\bar{\phi}_{N}\phi_{1}}&\ldots&\ldots&\ldots&\widehat{V}_{\bar{\phi}_{N}\phi_{N}}\end{array}\right],$
(89)

and ${\mathcal{H}}_{\chi}$ is as in Eq. (57), therefore the problem reduces to
analysing the $2\,N\times 2\,N$ ${\mathcal{H}}_{\phi}$ matrix, which in
principle is not block diagonal itself.

##  5 Application to Quintessence Models

Current data indicates that the universe is dominated by dark energy,
therefore if Eq. (1) is satisfied, this implies that this cannot be the result
of a positive cosmological constant nor can be described by a state at the
minimum of a potential with positive energy density. In such a case we can
invoke Quintessence models, which are described by rolling fields. It is known
that exponential potentials of the form $V=V_{0}\,e^{\lambda\,\chi}$ [45] and

$\displaystyle
V(\chi)=V_{1}\,e^{\lambda_{1}\,\chi}\,+\,V_{2}\,e^{\lambda_{1}\,\chi},$ (90)

with varying $\lambda_{i}$ parameters [49] that can fit well the constraints
based on Supernovae type Ia (SNeIa), Cosmic Microwave Background (CMB) and
Baryon Acoustic Oscillations (BAO) data (see [50] for an alternative
conclusion). For the case of one rolling field $\chi$, and hence only one term
in Eq. (90), $V(\chi)=V_{1}\,e^{\lambda\,\chi}$, the variable $\lambda$ is
constrained to be $\lambda=c$, which is the constant appearing in Eq. (1), and
must satisfy $\lambda\lesssim 0.6$. In [44] the mechanism that we use in this
work to satisfy the RdSC was applied to uplift a vacuum to a de Sitter vaccum
that could be in agreement with the RdSC but it was concluded that the
breaking of supersymmetry needed for the de Sitter uplift cannot be only
caused by the Quintessence field $\chi$ due to the requirement $\gamma=c>3$.
In the models we present in section 3 we do not have such a requirement and
hence even our minimal model with one rolling field plus one de Sitter field
could be used as a Quintessence model. In particular our result for the
minimal model for the constraint on $\gamma$ is compatible with a value of
$0.6$.

The potential with two terms, Eq. (90), is in fact a kind of freezing model
associated with scaling solutions [51], where the field equation of state
scales as that of the background fluid during most of the matter era. Here the
constants $\lambda_{i}$ are constrained to $\lambda_{1}\gg 1$ and
$\lambda_{2}\lesssim 1$ [49]. In the early matter era, the potential is
approximated by the first term in Eq. (90) while at late times the second
term. In [45] it was shown that the potential of Eq. (90) with
$\lambda\approx\lambda_{1}>>\sqrt{3}$ in the early universe and then switches
to $\lambda\approx\lambda_{2}=c=0.6$ at some recent point in the past.
Together these two stages approxiate the boundary trajectory that the dark
energy equation of state, $\omega(a)=\omega_{0}+\omega_{a}(1-a)$, where $a$ is
the scale factor normalized as $a=1$ today, needs to satisfy in order to be in
agreement with SNeIa, CMB and BAO and the RdSC, Eq. (1), as shown in [45]. The
No-Scale models that we study in section 3 can easily accommodate these
requirements. We are aware about the difficulties in constructing Quintessence
models in supergravity [52], where either it is difficult to evade
gravitational tests or having the Quintessence/supegravity models behave like
pure cosmological constants. Nevertheless, our result is encouraging from the
viewpoint of making compatible a supergravity theory with the Refined de
Sitter conjecture 333There is a current debate on the value of the Hubble
constant, [53, 54] which may lead to a revolution in cosmology. Solutions for
this tension may even discard Quintessence models, [55, 56]. Nevertheless, one
can obviously not rule out Quintessence models on this basis, as the model
realized in nature resolving the tension has not been established yet. In
addition, there may not even be a need for new physics as potential
reconsiderations of instrument calibration or data analysis may hold a key in
resolving the Hubble tension. For example, Cepheid calibration
reconsiderations can settle on a value in agreement with Standard Cosmology
[57]..

##  6 Conclusions

In this paper, we used the Refined de Sitter Conjecture (RdSC) and the Trans-
Planckian Censorship Conjecture (TCC) to constrain the parameter space of some
no-scale supergravity models. Specifically, we have studied a toy model of one
field with no-scale Kähler potential and a superpotential constructed from two
Minkowski endpoints, which yields a de Sitter scalar potential. It is well
known that for a particular choice of parameters, this model is unstable along
the imaginary direction, that is the second derivative of the scalar potential
is negative. But this is exactly the kind of model that the RdSC allows and
the TCC can constrain. To check the compatibility with the RdSC and the TCC we
first added a quartic term to the original Kähler potential to stabilize the
real part of the field, allowing us to work with one canonical field that is
proportional to the imaginary direction. Using this, we have shown that for
some choices of parameters the theory can be compatible with the RdSC and the
TCC. This analysis can be extended to the case of multi-field theories as well
as to cases with meta-stable de Sitter vacua, where the imaginary direction
can also be modified via a quartic term in the Kähler potential. We think this
kind of analysis is important as no-scale models with Minkowski/Anti-de Sitter
vacua generically appear as low energy limits of string compactifications, but
de Sitter vacua _“have not yet been rigorously shown to be realized in string
theory”_[30].

To construct the second class of models we presented in this paper, for which
an effective dS background is obtained from a rolling potential, we modified
the Kähler and superpotential of the simplest no-scale model we considered by
adding the terms corresponding to the rolling fields. We found that this
modification alleviates the instability and flatness along the imaginary and
the real directions without the need for quartic terms. The existence of the
rolling direction provides an escape from the swamp as the first criterion of
the RdSC can be satisfied for suitable values of parameters. Interestingly, we
found that the height of the potential is controlled by the no-scale
parameters while the rolling parameters make the model compatible with the
RdSC. These models could be used to construct viable cosmological models to
accommodate the accelerating expansion of the late Universe.

## Acknowledgements

L. V.-S. acknowledges the support from the “Fundamental Research Program” of
the Korea Institute for Advanced Study and the warm hospitality and
stimulating environment. M. T. is supported by the research deputy of Sharif
University of Technology. We are grateful to K. Kaneta, J. Kersten and K.
Olive for helpful discussions and J. Kersten and F. Borzumati for comments on
the manuscript.

##  Appendix A Canonical Normalization

To evaluate the second criterion of the RdSC, Eq. (2), we split the complex
field $\phi$ into real and imaginary components $\phi=x+iy$, to have

$\displaystyle{\mathcal{L}}=-K_{\phi\bar{\phi}}\,\partial_{\mu}\phi\,\partial^{\mu}\bar{\phi}=-\,\frac{3\alpha}{4\,[{\rm{Re}}[\phi]]^{2}}\,\left[(\partial_{\mu}x)(\partial^{\mu}x)+(\partial_{\mu}y)(\partial^{\mu}y)\right].$
(91)

With the Kähler metric of Eq. (23) we can fix the real part to b, such that

$\displaystyle\langle{\rm{Re}}[\phi]\rangle=b.$ (92)

Then we identify the canonical kinetic term of the fields $\chi$ and $\sigma$:

$\displaystyle{\mathcal{L}}=-\left(\frac{1}{\sqrt{2}}\partial_{\mu}\chi\right)^{2}-\left(\frac{1}{\sqrt{2}}\partial_{\mu}\sigma\right)^{2},$
(93)

we would have

$\displaystyle\frac{1}{\sqrt{2}}\,\partial_{\mu}\chi=\sqrt{\frac{3\alpha}{4\,[{\rm{Re}}[\phi]]^{2}}}\,\partial_{\mu}x\,,$
(94)
$\displaystyle\frac{1}{\sqrt{2}}\,\partial_{\mu}\sigma=\sqrt{\frac{3\alpha}{4\,}}\,\frac{\partial_{\mu}y}{b}\,\rightarrow\,\sigma=\sqrt{\frac{3\alpha}{2}}\,\frac{y}{b}.$
(95)

But since the real part is fixed and acquires a vev, then our effective theory
will consist in only one field, $\sigma$, with canonical kinetic term
${\mathcal{L}}=-\frac{1}{2}\,\partial_{\mu}\sigma\partial^{\mu}\sigma$.

##  Appendix B Details of Stability Conditions

For the _2+1 Model_ the second derivatives of $\widehat{V}$ are as follows

$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}^{1}_{\;\;1}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}_{1}^{\;1}=\frac{4\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}(3r_{2}^{2}\alpha_{1}^{2}-2r_{1}^{2}\alpha_{2}+3\,\gamma\alpha_{1}\alpha_{2}r_{1}^{2})}{s^{2}\alpha_{1}^{2}\alpha_{2}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}^{11}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}_{11}=\frac{4\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}(s^{2}\,(2\,r_{1}^{2}\alpha_{2}-9\,r_{1}^{2}\alpha_{1}\alpha_{2}-3r_{2}^{2}\alpha_{1}^{2})+9\,r_{1}^{2}(r_{2}^{2}\alpha_{1}+r_{1}^{2}\alpha_{2}))}{s^{4}\alpha_{1}^{2}\alpha_{2}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}^{1}_{\;\;2}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}_{1}^{\;2}=\frac{12\,\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}r_{1}r_{2}(\gamma-1)\phi_{1}}{s^{2}\,\alpha_{1}\phi_{2}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}^{12}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{1}_{1}\widehat{V}_{12}=\frac{12\,\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}\,r_{1}r_{2}(3r_{1}^{2}\alpha_{2}+3r_{2}^{2}\alpha_{1}-2\,s^{2}\alpha_{1}\alpha_{2})\phi_{1}}{s^{4}\alpha_{1}^{2}\alpha_{2}\phi_{2}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}^{2}_{\;\;1}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}_{2}^{\;1}=\frac{12\,\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}r_{1}r_{2}(\gamma-1)\phi_{2}}{s^{2}\,\alpha_{2}\phi_{1}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}^{21}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}_{21}=\frac{12\,\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}\,r_{1}r_{2}(3r_{1}^{2}\alpha_{2}+3r_{2}^{2}\alpha_{1}-2\,s^{2}\alpha_{1}\alpha_{2})\phi_{2}}{s^{4}\alpha_{2}^{2}\alpha_{1}\phi_{1}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}^{22}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}_{22}=\frac{4\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}(s^{2}\,(2\,r_{2}^{2}\alpha_{1}-9\,r_{2}^{2}\alpha_{2}\alpha_{1}-3r_{1}^{2}\alpha_{2}^{2})+9\,r_{2}^{2}(r_{1}^{2}\alpha_{2}+r_{2}^{2}\alpha_{1}))}{s^{4}\alpha_{2}^{2}\alpha_{1}},$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}^{2}_{\;\;2}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{\widetilde{G}}}K^{2}_{2}\widehat{V}_{2}^{\;2}=\frac{4\,\times
2^{-3\,(\alpha_{1}+\alpha_{2})}a^{2}(3r_{1}^{2}\alpha_{2}^{2}-2r_{2}^{2}\alpha_{1}+3\,\gamma\alpha_{2}\alpha_{1}r_{2}^{2})}{s^{2}\alpha_{2}^{2}\alpha_{1}}.$
(96)

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