Bounds on tower mass scales in the presence of throats of different warping

Min-Seok Seoa

aDepartment of Physics Education, Korea National University of Education,
Cheongju 28173, Republic of Korea

In Type IIB flux compactification realizing the metastable de Sitter (dS)
vacuum, the uplift potential can be generated by $\overline{\rm D3}$-branes at
the tip of Klebanov-Strassler throat. Then the uplift potential obeys the
scaling law with respect to the tower mass scale $m_{\rm sc}$, which can be
the Kaluza-Klein (KK) mass scale associated with the throat containing
$\overline{\rm D3}$-branes or the bulk tower mass scales, depending on the
warping of the throat. On the other hand, in the presence of another throat of
stronger warping, the KK mass scale associated with this throat is lower than
$m_{\rm sc}$. Nevertheless, the Higuchi bound and the condition that the tower
mass scale is higher than the gravitino mass provide the upper bound on
$m_{\rm sc}$ determined by the lowest tower mass scale (or gravitino mass).
This bound also can be interpreted as the lower bound on the lowest tower mass
scale determined by $m_{\rm sc}$. We investigate this bound in detail when the
throat containing $\overline{\rm D3}$-branes is strongly and weakly warped,
respectively.

##  1 Introduction

The swampland program [2] has provided a set of conjectured constraints that
the low energy effective field theory (EFT) must satisfy in order to have a UV
completion in quantum gravity (for reviews, see, e.g., [3, 4, 5, 6, 7, 8]).
Among various proposals in the program, the instability of de Sitter (dS)
space formulated by the dS swampland conjecture is of particular interest [9,
10, 11] (see also [12, 13, 14]) as string realization of the metastable dS
vacuum [15, 16] requires a tuning between several ingredients such as flux
compactification, non-perturbative effect, and uplift, which has led to the
debate on the consistency of the model. In the justification of the dS
swampland conjecture the distance conjecture plays the crucial role [14]. It
states that the infinite distance limit of the moduli space corresponds to the
corner of the landscape, at which the EFT becomes invalid as the mass scale of
a tower of states decreases rapidly [17]. Such a descent of a tower of states
implies the rapid increase in the number of low energy degrees of freedom
hence their production in dS space can violate the covariant entropy bound
[18] given by (horizon area)$/4$ (see also [19, 20, 21, 22, 23]).

The dS swampland conjecture raised the suspicion that our universe well
described by a positive cosmological constant, $\Lambda=3m_{\rm Pl}^{2}H^{2}$,
where $H$ is the Hubble parameter (the inverse of the horizon radius), is
close to the swampland in the moduli space. In this regard, there have been
attempts to formulate the closeness of our universe to the swampland in the
form of the scaling law, reflecting the distance conjecture. The anti-
dS(AdS)/dS distance conjecture focuses on the smallness of $\Lambda\sim
10^{-123}m_{\rm Pl}^{4}$ and suggests the relation $m\sim|\Lambda|^{\alpha}$,
where $m$ is some tower mass scale [24]. If $m$ is the Kaluza-Klein (KK) mass
scale, the lower bound on $\alpha$ is given by $1/4$ [25], which can be
obtained from the observational bound on the size of extra dimensions [26,
27]. Moreover, any nonzero mass of the state in dS space is required to be
larger than the Higuchi bound [28] (see [29] for a review) given by
$\sqrt{s(s-1)}H\sim\Lambda^{1/2}$, where $s$ is the spin of the state,
indicating that $\alpha$ is smaller than $1/2$. On the other hand, in the
effective supergravity description of string theory, $|\Lambda|$ of the AdS
vacuum (for the metastable dS vacuum, the size of the AdS vacuum energy
density before uplift which will be denoted by $V_{\rm AdS}$) is smaller than
$3m_{\rm Pl}^{2}m_{3/2}^{2}$, where $m_{3/2}$ is the gravitino mass and the
inequality is saturated in the supersymmetric case. The gravitino distance
conjecture claims that it may be $m_{3/2}$ rather than $|\Lambda|$ which obeys
the scaling law [30, 31, 32].

In the string model realizing the metastable dS vacuum based on Type IIB flux
compactification, when the uplift potential $V_{\rm up}$ is generated by
$\overline{\rm D3}$-branes at the tip of the Klebanov-Strassler throat [33],
it turns out that the size of $V_{\rm up}$ obeys the scaling law with respect
to the tower mass scale [34]. If the throat containing $\overline{\rm
D3}$-branes is strongly warped, the mass scale of the KK modes localized in
this throat region satisfies $m_{\rm KK}^{\rm throat}\sim V_{\rm up}^{1/4}$
[35]. We note that in the presence of a number of throats [36], the KK mass
scale associated with the throat of the strongest warping is the lowest tower
mass scale. Thus, if the warping of the throat containing $\overline{\rm
D3}$-branes is the strongest, the scaling law relates $V_{\rm up}$ and the
lowest tower mass scale. Moreover, the exponent $1/4$ is nothing more than the
inverse of the number of noncompact dimensions over which $\overline{\rm
D3}$-branes are extended. In contrast, when the warping is extremely weak,
both the bulk tower mass scales and $V_{\rm up}$ scale with respect to the
size of internal volume so we can find the scaling law between them [34]. More
concretely, the string scale $m_{s}$ satisfies $m_{s}\sim V_{\rm up}^{1/4}$
where the exponent is the inverse of the number of noncompact dimensions. In
addition, the bulk KK mass scale, the lowest tower mass scale in the absence
of the throat of stronger warping, also obeys the scaling law but the exponent
in this case is given by $1/3$ : $m_{\rm KK}^{\rm bulk}\sim V_{\rm up}^{1/3}$.

Since the uplift potential given by $V_{\rm up}=\Lambda+|V_{\rm AdS}|$ is
connected to the tower mass scale $m_{\rm sc}$ through the scaling law $m_{\rm
sc}\sim V_{\rm up}^{\alpha}$ with $\alpha=1/4$ or $1/3$, we find the
inequality $m_{\rm sc}>\Lambda^{\alpha}$ for the positive $\Lambda$. Hence,
unlike the AdS/dS distance conjecture suggesting the equality, $m_{\rm sc}$
may not be as light as $\sim\Lambda^{\alpha}$. Meanwhile, if there exists a
throat whose warping is stronger than that of the throat containing
$\overline{\rm D3}$-branes, the KK mass scale associated with it is lower than
$m_{\rm sc}$, the tower mass scale satisfying the scaling law with respect to
$V_{\rm up}$. Nevertheless, for the EFT based on the four-dimensional
supergravity description to be valid, the lowest tower mass scale is required
to be larger than $m_{3/2}$. Combining this with the Higuchi bound and the
fact that $|V_{\rm AdS}|$ is smaller than $3m_{\rm Pl}^{2}m_{3/2}^{2}$, we
obtain the inequality obeyed by the lowest tower mass scale as well as $m_{\rm
sc}$. This will be explored in detail in this article. As we will see, in the
presence of the lower tower mass scale, $m_{\rm sc}$ has the upper bound
determined by the lower tower mass scale, or equivalently, the lowest tower
mass scale has the lower bound determined by $m_{\rm sc}$. We expect that our
results may be useful in the phenomenological study on the structure of extra
dimensions.

This article is organized as follows. Section 2 is devoted to the brief
reviews on the Higuchi bound and the connection between tower mass scales and
$V_{\rm up}$, which provide the background for our discussion. Based on them,
in Section 3, we obtain the upper bound on $m_{\rm sc}$ in terms of the lower
tower mass scale when the throat containing $\overline{\rm D3}$-branes is
strongly and weakly warped, respectively. Then we conclude.

##  2 Reviews on Higuchi bound, tower mass scale and uplift

###  2.1 Higuchi bound

We begin our review on the Higuchi bound with the discussions on the unitary
irreducible representations (UIRs) of SO(1,4) dS isometry group and their
masses. For details, we refer the reader to [29] and references therein. The
‘mass’ of the state in the UIR is determined by the quadratic Casimir,

$\begin{split}Q=-\frac{1}{2}L_{AB}L^{AB},\end{split}$ (1)

where $L_{AB}$ ($A,B=0,1,\cdots,4$) are SO(1,4) generators, as the field
equation of motion is reduced to its eigenvalue equation,

$\begin{split}\Big{(}Q-\langle Q\rangle\Big{)}{\cal K}(x)=0.\end{split}$ (2)

Here the field ${\cal K}(x)$ carries the tensor or spinor indices
corresponding to the UIR to which the field belongs. The eigenvalue of $Q$ is
given by [37] (see also Section 12 of [29])

$\begin{split}\langle Q\rangle=-s(s+1)-(q+1)(q-2),\end{split}$ (3)

where $s$ is interpreted as a spin in the $H\to 0$ limit and $q\in\mathbb{C}$
is determined by the type of the representation. 111A pair of numbers $(s,q)$
labelling the UIR can be obtained by observing the representation of SO(4)
isometry subgroup. Since SO(4)$\cong$ SU(2)$\times$SU(2) and the quadratic
Casimirs of two SU(2)s have eigenvalues $j_{\ell}(j_{\ell}+1)$ and
$j_{r}(j_{r}+1)$ ($j_{\ell},j_{r}\in\mathbb{Z}/2$), respectively, the
irreducible representation of SO(4) is characterized by $(j_{\ell},j_{r})$.
Then $s$ is the infimum (greatest lower bound) of $j_{\ell}+j_{r}$, which is
interpreted as a spin in the $H\to 0$ limit. On the other hand, SO(1,4)/SO(4)
generators raise/lower the $j_{\ell}$ and $j_{r}$ values, and their
contributions to the SO(1,4) quadratic Casimir $Q$ determine the value of $q$.
For example, in the discrete series of representations, we can find two dual
representations in which the values of $({\rm max}(j_{r}-j_{\ell}),{\rm
min}(j_{r}-j_{\ell}))$ for the states satisfying $j_{r}+j_{\ell}=s$ are given
by $(q,s)$ and $(-s,-q)$, respectively.  The types of SO(1,4) UIRs are
classified in [37], and it turns out that under the Poincaré contraction,
i.e., in the $H\to 0$ limit, a particular set of UIRs can be reduced to the
wavefunction in Minkowski space in a sensible way (the positive and negative
frequency modes are well separated) [38] (see also Section 14 of [29]) :

  * •

The principal series of representations : In this case, $q=\frac{1}{2}+i\nu$,
where $\nu\geq 0$ for an integer $s$ and $\nu>0$ for a half-integer $s$, such
that

$\begin{split}\langle Q\rangle=-s(s+1)+\frac{9}{4}+\nu^{2}.\end{split}$ (4)

The representation of this type is also called the massive representation
since in the $H\to 0$ limit, $\nu H$ is reduced to the mass of the field.

  * •

The complementary series of representations : The value of $s$ in this case is
only an integer. Moreover, $q=\frac{1}{2}+\nu$, where $\nu\in\mathbb{R}$ and

$\begin{split}&0<|\nu|<\frac{3}{2}\quad\quad{\rm for}\quad s=0,\\\
&0<|\nu|<\frac{1}{2}\quad\quad{\rm for}\quad s=1,2,3,\cdots,\end{split}$ (5)

giving

$\begin{split}\langle Q\rangle=-s(s+1)+\frac{9}{4}-\nu^{2}.\end{split}$ (6)

The $H\to 0$ limit of the representation of this type is sensible when $s=0$
and $\nu=1/2$ (thus $q=1$), which corresponds to the conformally coupled
massless spin-$0$ field.

  * •

The discrete series of representations : In this case, $q=0,1,\cdots,s-1,s$
for an integer $s$ and $q=\frac{1}{2},\cdots,s-1,s$ for a half-integer $s$.
The sensible representation in the $H\to 0$ limit requires that $s=q>0$, which
is reduced to the massless spin-$s$ field.

Moreover, the quadratic Casimir of dS isometry is not exactly identified with
the Laplacian : the quadratic Casimir eigenvalue equation given by (2) is
written as [39]

$\begin{split}\big{(}\square-H^{2}s(s+2)-H^{2}\langle Q\rangle\big{)}{\cal
K}(x)=0.\end{split}$ (7)

On the other hand, from the fact that the representation belonging to the
discrete series satisfying $s=q>0$ becomes the massless spin-$s$ field in the
$H\to 0$ limit, it was suggested to define the ‘mass’ of the state in dS space
by [39]

$\begin{split}m^{2}=H^{2}\big{(}\langle Q\rangle-\langle
Q_{s=q}\rangle\big{)}=H^{2}\big{(}\langle
Q\rangle+2(s^{2}-1)\big{)}=(s-q)(s+q-1)H^{2},\end{split}$ (8)

such that $m^{2}=0$ for $s=q>0$. For the representations in the principal
series, the mass defined in this way can be written as
$m^{2}=\big{(}s-\frac{1}{2}\big{)}^{2}H^{2}+\nu^{2}H^{2}$ (note that
$q=\frac{1}{2}+i\nu$), so for a finite value of $s$, $m^{2}$ is reduced to
$\nu^{2}H^{2}$ in the $H\to 0$ limit. In terms of $m$, (2) becomes

$\begin{split}\big{(}\square-(2-s(s-2))H^{2}-m^{2}\big{)}{\cal
K}(x)=0,\end{split}$ (9)

which indeed coincides with the equation of motion used in the previous
literatures, e.g., [28, 40, 41, 42].

It is remarkable that apart from the issue of the sensible $H\to 0$ limit,
regarding (8) as a mass of any state in the UIR, one finds that the nonzero
value of mass has a lower bound called the Higuchi bound. More concretely, the
value of $m^{2}$ turns out to be either zero or larger than $s(s-1)H^{2}$,
which is meaningful for $s>1$. This bound is saturated by the representations
belonging to the complementary series with $\nu=\pm\frac{1}{2}$ ($q=1,0$) and
the discrete series with $q=0$. For instance, for $s=2$, the lower bound on
the nonzero value of $m^{2}$ is given by $2H^{2}$ [28] (see also [40] for
$s>2$).

###  2.2 Tower mass scales in the presence of throat and uplift potential

Throughout this article, we consider Type IIB Calabi-Yau orientifold
compactifications containing a number of Klebanov-Strassler throats, in which
the dilaton and complex structure moduli are stabilized by fluxes [43]. The
Kähler moduli are stabilized by non-perturbative effect [15] and possibly the
additional $\alpha^{\prime}$ corrections [16], and the potential stabilizing
all the moduli is uplifted by $\overline{\rm D3}$-branes at the tip of one of
throats. The string scale $m_{s}$, the mass scale of a tower of string
excitations is given by $1/(2\pi\sqrt{\alpha^{\prime}})$. Since the ten-
dimensional gravitational coupling is given by
$\kappa_{10}^{2}=g_{s}^{2}/(4\pi m_{s}^{8})$, denoting the volume of the
internal manifold by ${\cal V}/m_{s}^{6}$, we obtain the relation between
$m_{s}$ and the four-dimensional Planck scale $m_{\rm Pl}$ :

$\begin{split}m_{s}=\frac{g_{s}}{\sqrt{4\pi{\cal V}}}m_{\rm Pl}.\end{split}$
(10)

Moreover, under the compactification, there can be various KK mass scales
depending on where the KK modes are localized. The mass scale of the KK modes
in the bulk is given by

$\begin{split}m_{\rm KK}^{\rm bulk}=\frac{2\pi m_{s}}{{\cal
V}^{1/6}}=\sqrt{\pi}\frac{g_{s}}{{\cal V}^{2/3}}m_{\rm Pl}.\end{split}$ (11)

On the other hand, the mass scale of the KK modes localized in the throat
region is determined by how strong the warping of the throat is. To see this,
we note that the metric near the tip of the throat is given by

$\begin{split}ds^{2}=e^{2\Omega_{4}(x,y)}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2\Omega_{6}(x,y)}g_{mn}dx^{m}dx^{n},\end{split}$
(12)

where

$\begin{split}e^{2\Omega_{4}(x,y)}=e^{2A(y)}e^{2\Omega(x)},\quad\quad
e^{2\Omega_{6}(x,y)}=e^{-2A(y)}\sigma(x)^{1/2}.\end{split}$ (13)

This throat geometry is supported by fluxes of $F_{3}$ and $H_{3}$, the flux
quanta of which in string units will be denoted by $M$ and $K$, respectively.
In the metric, $\sigma(x)$ is the scalar part of the volume modulus, the
vacuum expectation value of which satisfies ${\cal
V}=\langle\sigma^{3/2}\rangle$ under the normalization $\int
d^{6}y\sqrt{g_{6}}e^{-4A}=m_{s}^{-6}$. Then the Weyl factor $e^{2\Omega(x)}$
can be written as

$\begin{split}e^{2\Omega(x)}=\frac{{\cal V}\ell_{s}^{6}}{\sigma(x)^{3/2}\int
d^{6}y\sqrt{g_{6}}e^{-4A}}=\frac{\langle\sigma^{3/2}\rangle}{\sigma(x)^{3/2}},\end{split}$
(14)

such that $\langle e^{2\Omega(x)}\rangle=1$. The warping of the throat is
typically parametrized by the ‘warp factor’ defined by $e^{-4A}$, which can be
written as

$\begin{split}e^{-4A(y)}=1+\frac{e^{-4A_{0}(y)}}{\sigma(x)},\end{split}$ (15)

where

$\begin{split}&e^{-4A_{0}(y)}=2^{2/3}\frac{(g_{s}M)^{2}}{(2\pi)^{4}|z|^{4/3}}I(\eta),\\\
&I(\eta)=\int_{\eta}^{\infty}dx\frac{x\coth
x-1}{\sinh^{2}x}(\sinh(2x)-2x)^{1/3}.\end{split}$ (16)

Then the throat KK mass scale is given by $m_{\rm KK}^{\rm throat}=\langle
e^{\Omega_{4}}\rangle/(\langle e^{\Omega_{6}}\rangle R)=\langle
e^{2A}\rangle/(R{\cal V}^{1/6})$, where $R$ is the typical length scale of the
throat : $R\sim|z|^{1/3}/m_{s}$ for the A-cycle and $R\sim\eta_{\rm
UV}|z|^{1/3}/m_{s}$ for the B-cycle, where $\eta_{\rm
UV}\sim\log\big{(}\frac{1}{|z|}\big{)}=\frac{2\pi}{g_{s}}\frac{K}{M}$($>1$) is
the length of the throat. When the throat is strongly warped, $e^{-4A}\gg 1$
is satisfied, and the throat KK mass scale is highly redshifted by the warp
factor [44] (see also [45, 46] for recent discussions) :

$\begin{split}m_{\rm KK}^{\rm
throat}=\frac{2^{1/2}3^{1/6}\pi^{3/2}}{I(0)^{1/2}}\frac{|z|^{1/3}}{M{\cal
V}^{1/3}}m_{\rm Pl}.\end{split}$ (17)

Moreover, for the KK modes localized along the B-cycle of the throat, the
corresponding KK mass scale is additionally suppressed by $\eta_{\rm UV}$
[45]. Therefore, the KK mass scale associated with the throat of the strongest
warping (that is, the smallest $|z|$ and the largest $\eta_{\rm UV}$) is
typically the lowest tower mass scale. In contrast, when the throat is
extremely weakly warped, i.e., $e^{-4A}\simeq 1$, the throat KK mass scale is
given by

$\begin{split}m_{\rm KK}^{\rm throat}=\frac{g_{s}}{|z|^{1/3}{\cal
V}^{2/3}}m_{\rm Pl}.\end{split}$ (18)

Comparing this with (11), one finds that for the extremely weakly warped
throat, the bulk KK mass scale $m_{\rm KK}^{\rm bulk}$ is the lowest tower
mass scale.

Now we move onto the uplift potential $V_{\rm up}$. When ${\overline{\rm
D3}}$-branes at the tip of the throat are extended over the noncompact four-
dimensional spacetime, the induced metric is given by

$\begin{split}ds_{{\overline{\rm
D3}}}^{2}=e^{2\Omega_{4}(x,y)}g_{\mu\nu}dx^{\mu}dx^{\nu}=e^{2A(y)}e^{2\Omega(x)}g_{\mu\nu}dx^{\mu}dx^{\nu},\end{split}$
(19)

from which $V_{\rm up}$ is written as

$\begin{split}V_{\rm up}=2p\frac{T_{3}}{g_{s}}e^{4\Omega_{4}(x,y)}=4\pi
p\frac{m_{s}^{4}}{g_{s}}e^{4A(y)}e^{4\Omega(x)},\end{split}$ (20)

where $T_{3}=2\pi m_{s}^{4}$ is the brane tension and $p$ is the number of
$\overline{\rm D3}$-branes. For the strongly warped throat ($e^{-4A}\gg 1$),
we obtain

$\begin{split}V_{\rm
up}=\frac{2^{4/3}\pi^{3}}{I(0)}\frac{g_{s}p}{M^{2}}\frac{|z|^{4/3}}{{\cal
V}^{4/3}}m_{\rm Pl}^{4}.\end{split}$ (21)

Comparing this with (17), one finds the scaling law,

$\begin{split}m_{\rm KK}^{\rm
throat}\sim\frac{1}{g_{s}^{1/4}M^{1/2}p^{1/4}}\langle V_{\rm
up}\rangle^{1/4},\end{split}$ (22)

where the exponent $1/4$ is the inverse of the number of noncompact dimensions
over which $\overline{\rm D3}$-branes are extended. Meanwhile, when the throat
is weakly warped ($e^{-4A}\simeq 1$), the uplift potential is given by

$\begin{split}V_{\rm up}=\frac{g_{s}^{3}}{4\pi}\frac{p}{{\cal V}^{2}}m_{\rm
Pl}^{4},\end{split}$ (23)

from which one finds two scaling laws with respect to $m_{s}$ and $m_{\rm
KK}^{\rm bulk}$ given by (10) and (11), respectively :

$\begin{split}&m_{s}\sim\Big{(}\frac{g_{s}}{4\pi p}\Big{)}^{1/4}\langle V_{\rm
up}\rangle^{1/4},\\\ &m_{\rm KK}^{\rm
bulk}\sim\frac{1}{p^{1/3}}\Big{\langle}\frac{V_{\rm up}}{m_{\rm
Pl}^{4}}\Big{\rangle}^{1/3}m_{\rm Pl}.\end{split}$ (24)

We note that the exponent in the scaling law with respect to $m_{s}$ is $1/4$,
the inverse of the number of noncompact dimensions. Moreover, $m_{\rm KK}^{\rm
bulk}$ is the lowest tower mass scale.

##  3 Relation between two tower mass scales

In string model, the metastable dS vacuum is realized by uplift of the AdS
vacuum, indicating that the positive cosmological constant $\Lambda$ can be
written as

$\begin{split}\Lambda=3m_{\rm Pl}^{2}H^{2}=-|V_{\rm AdS}|+V_{\rm
up}.\end{split}$ (25)

As reviewed in the previous section, $V_{\rm up}$ obeys the scaling law with
respect to some tower mass scale which will be denoted by $m_{\rm sc}$,

$\begin{split}V_{\rm up}=v_{0}m_{\rm Pl}^{4}\Big{(}\frac{m_{\rm sc}}{m_{\rm
Pl}}\Big{)}^{1/\alpha}.\end{split}$ (26)

For the strongly warped throat, $m_{\rm sc}$ is identified with $m_{\rm
KK}^{\rm throat}$ given by (17) and $\alpha=1/4$. For the extremely weakly
warped throat, $m_{\rm sc}$ corresponds to $m_{s}$ given by (10)
($\alpha=1/4$) or $m_{\rm KK}^{\rm bulk}$ given by (11) ($\alpha=1/3$).

We now consider another throat of stronger warping such that the mass scale
$m_{0}$ of KK modes localized in this throat is lower than $m_{\rm sc}$.
Denoting the conifold modulus and the flux quanta of $F_{3}$ associated with
the throat of the stronger warping by $z_{\ell}$ and $M_{\ell}$, respectively,
we obtain

$\begin{split}m_{0}\simeq\frac{|z_{\ell}|^{1/3}}{M_{\ell}{\cal V}^{1/3}}m_{\rm
Pl},\end{split}$ (27)

just like (17). The Higuchi bound imposes that the nonzero $m_{0}$ is larger
than $\sqrt{s(s-1)}H$, or equivalently, $H<m_{0}/\sqrt{s(s-1)}$. Combining
this with the fact that $|V_{\rm Ads}|$ is smaller than $3m_{\rm
Pl}^{2}m_{3/2}^{2}$, (25) provides the inequality

$\begin{split}3m_{\rm Pl}^{2}m_{3/2}^{2}>|V_{\rm AdS}|>v_{0}m_{\rm
Pl}^{4}\Big{(}\frac{m_{\rm sc}}{m_{\rm
Pl}}\Big{)}^{1/\alpha}-\frac{3}{s(s-1)}m_{\rm Pl}^{2}m_{0}^{2},\end{split}$
(28)

which relates three scales, $m_{\rm sc}$, $m_{0}$, and $m_{3/2}$. Noting that
$m_{3/2}$ is the characteristic mass scale of the four dimensional
supergravity formalism, we expect that this is lower than $m_{0}$, the KK mass
scale implying the existence of extra dimensions, hence $3m_{\rm
Pl}^{2}m_{0}^{2}>3m_{\rm Pl}^{2}m_{3/2}^{2}$. Then we obtain the inequality

$\begin{split}3\Big{(}1+\frac{1}{s(s-1)}\Big{)}m_{\rm
Pl}^{2}m_{0}^{2}>v_{0}m_{\rm Pl}^{4}\Big{(}\frac{m_{\rm sc}}{m_{\rm
Pl}}\Big{)}^{1/\alpha},\end{split}$ (29)

which indicates that $m_{\rm sc}$ ($m_{0}$) has the upper (lower) bound
determined by $m_{0}$ ($m_{\rm sc}$). We investigate this inequality in detail
when the throat containing $\overline{\rm D3}$-branes is strongly and weakly
warped, respectively.

###  3.1 Strong warping case ($e^{-4A}\gg 1$)

When the throat containing $\overline{\rm D3}$-branes is strongly warped, the
scaling law (22) is satisfied, indicating $m_{\rm sc}=m_{\rm KK}^{\rm throat}$
(given by (17)), $\alpha=1/4$, and $v_{0}\simeq g_{s}pM^{2}$. Then (28) is
written as

$\begin{split}3m_{\rm Pl}^{2}m_{3/2}^{2}>|V_{\rm AdS}|>(g_{s}pM^{2})(m_{\rm
KK}^{\rm throat})^{4}-\frac{3}{s(s-1)}m_{\rm Pl}^{2}m_{0}^{2},\end{split}$
(30)

and (29) becomes

$\begin{split}m_{\rm KK}^{\rm
throat}<\Big{(}\frac{3(s^{2}-s+1)}{(g_{s}pM^{2})s(s-1)}\Big{)}^{1/4}m_{\rm
Pl}^{1/2}m_{0}^{1/2},\end{split}$ (31)

or equivalently,

$\begin{split}\Big{(}\frac{m_{0}}{m_{\rm KK}^{\rm
throat}}\Big{)}^{2}>(g_{s}pM^{2})\frac{s(s-1)}{3(s^{2}-s+1)}\Big{(}\frac{m_{\rm
KK}^{\rm throat}}{m_{\rm Pl}}\Big{)}^{2}.\end{split}$ (32)

We may rewrite this bound in terms of the explicit expressions (17) for
$m_{\rm KK}^{\rm throat}$ and (27) for $m_{0}$ to obtain the constraint on the
warping of throats :

$\begin{split}\Big{(}\frac{|z_{\ell}|}{|z|}\Big{)}^{1/3}\frac{M}{M_{\ell}}>\Big{(}\frac{(g_{s}pM^{2})s(s-1)}{3(s^{2}-s+1)}\Big{)}^{1/2}\frac{|z|^{1/3}}{M{\cal
V}^{1/3}}.\end{split}$ (33)

The bound (31) shows that even if the throat containing $\overline{\rm
D3}$-branes is not of the strongest warping so the associated KK mass scale is
not the lowest tower mass scale, it cannot be arbitrarily higher than the
lowest KK mass scale $m_{0}$, but bounded by $\lesssim m_{\rm
Pl}^{1/2}m_{0}^{1/2}$. This in turn means that $m_{0}$ cannot be arbitrarily
small, but has the lower bound determined by $m_{\rm KK}^{\rm throat}$.
Indeed, the size of deformation at the tip of the throat is given by
$(g_{s}M)^{1/2}/(2\pi m_{s})$, 222This comes from the fact that for the
strongly warped throat, $e^{-2A}$ is approximated by
$e^{-2A_{0}}/\sigma^{1/2}$ (see (15)). As can be inferred from (16), it
depends on $|z|$ and $\sigma$ through the combination
$1/(|z|^{2/3}\sigma^{1/2})$, which is cancelled by the prefactor
$\sigma^{1/2}$ in $G_{mn}=e^{-2A}\sigma^{1/2}g_{mn}$ and the overall factor
$|z|^{2/3}$ in $g_{mn}$ [47]. As a result, the overall factor in $G_{mn}$ is
given by $(g_{s}M)/(2\pi m_{s})^{2}$.  which is required to be much larger
than the string length $m_{s}^{-1}$ for the effective supergravity description
to be valid. This indicates that $g_{s}M\gg 1$. Moreover, we can further
impose $g_{s}M^{2}\gg p$ because otherwise the conifold modulus $z$ is
stabilized at $0$ [48] (see, however, [49] for a recent counterargument). Both
of these constraints impose that $g_{s}pM^{2}\gg 1$, so our bound is more or
less stronger than the simple inequality $m_{\rm KK}^{\rm throat}<{\cal
O}(1)m_{\rm Pl}^{1/2}m_{0}^{1/2}$.

In fact, (31) as the upper bound on $m_{\rm KK}^{\rm throat}$ is useful when
the size of $\Lambda$ is not negligibly small, which may be realized in the
inflationary cosmology. In contrast, $\Lambda$ in our universe is as small as
$10^{-123}m_{\rm Pl}^{4}$ so it is reasonable to take $|V_{\rm up}|$ to be
much larger than $\Lambda$. Since $V_{\rm up}\simeq|V_{\rm AdS}|$ in this
case, the condition $|V_{\rm AdS}|<3m_{\rm Pl}^{2}m_{3/2}^{2}$ gives a more
stringent bound,

$\begin{split}m_{\rm KK}^{\rm
throat}\lesssim\Big{(}\frac{3}{g_{s}pM^{2}}\Big{)}^{1/4}m_{\rm
Pl}^{1/2}m_{3/2}^{1/2}.\end{split}$ (34)

But still, the bound (31) is valid as the lower bound on $m_{0}$ determined by
$m_{\rm KK}^{\rm throat}$. In any case, it is remarkable that some
intermediate new physics scale has the upper bound determined by another much
lower scale.

We can also compare the bound (31) with the bound obtained from the species
scale $\Lambda_{\rm sp}$ [50, 51],

$\begin{split}\Lambda_{\rm sp}=\frac{m_{\rm Pl}}{\sqrt{N_{\rm
tot}}},\end{split}$ (35)

above which gravity is no longer weakly coupled to matter. Here $N_{\rm tot}$
is the number of low energy degrees of freedom below $\Lambda_{\rm sp}$, hence
given by

$\begin{split}&N_{\rm tot}=N_{0}+N_{\rm KK},\\\ &N_{0}=\frac{\Lambda_{\rm
sp}}{m_{0}},\quad\quad N_{\rm KK}=\frac{\Lambda_{\rm sp}}{m_{\rm
KK}}.\end{split}$ (36)

For $m_{0}\ll m_{\rm KK}$, we have $N_{0}\gg N_{\rm KK}$ then $\Lambda_{\rm
sp}\simeq m_{\rm Pl}/\sqrt{N_{0}}$, from which we obtain $\Lambda_{\rm
sp}\simeq m_{\rm Pl}^{2/3}m_{0}^{1/3}$. Since $m_{\rm KK}<\Lambda_{\rm sp}$, a
condition $m_{\rm KK}<m_{\rm Pl}^{2/3}m_{0}^{1/3}$ is satisfied, but this is
less stringent than (31).

###  3.2 Weak warping case ($e^{-4A}\simeq 1$)

When the throat containing $\overline{\rm D3}$-branes is extremely weakly
warped, two scaling laws given by (24) are satisfied : for $m_{\rm sc}=m_{\rm
KK}^{\rm bulk}$ (given by (11)) $v_{0}\simeq p$ and $\alpha=1/3$ while for
$m_{\rm sc}=m_{s}$ (given by (10)) $v_{0}\simeq(4\pi p)/g_{s}$ and
$\alpha=1/4$.

We first consider the case $m_{\rm sc}=m_{\rm KK}^{\rm bulk}$, in which the
inequality (28) is written as

$\begin{split}3m_{\rm Pl}^{2}m_{3/2}^{2}>|V_{\rm AdS}|>pm_{\rm Pl}(m_{\rm
KK}^{\rm bulk})^{3}-\frac{3}{s(s-1)}m_{\rm Pl}^{2}m_{0}^{2}.\end{split}$ (37)

In the presence of an additional throat which is strongly warped, the KK modes
localized in this throat provide the lowest tower mass scale given by (27).
Requiring $m_{0}>m_{3/2}$, we obtain

$\begin{split}3\Big{(}1+\frac{1}{s(s-1)}\Big{)}m_{\rm Pl}^{2}m_{0}^{2}>pm_{\rm
Pl}(m_{\rm KK}^{\rm bulk})^{3},\end{split}$ (38)

or equivalently,

$\begin{split}m_{\rm KK}^{\rm
bulk}<\Big{(}\frac{3(s^{2}-s+1)}{ps(s-1)}\Big{)}^{1/3}m_{\rm
Pl}^{1/3}m_{0}^{2/3}.\end{split}$ (39)

Thus, $m_{\rm KK}^{\rm bulk}$ has the upper bound depending on $m_{0}$, with
the exponent given by $2/3$. At the same time, this inequality may be
interpreted as the lower bound on $m_{0}$ as well. Putting the explicit
expressions for $m_{0}$ and $m_{\rm KK}^{\rm bulk}$ into the inequality gives
the constraint on the strongest warp factor :

$\begin{split}\frac{3(s^{2}-s+1)}{ps(s-1)}\Big{(}\frac{|z_{\ell}|^{1/3}}{M_{\ell}}\Big{)}^{2}>\frac{g_{s}^{3}}{{\cal
V}^{4/3}}.\end{split}$ (40)

Just like the previous case in which the throat containing $\overline{\rm
D3}$-branes is strongly warped, we have a more stringent upper bound on
$m_{\rm KK}^{\rm bulk}$ determined by $m_{3/2}$ if $|V_{\rm AdS}|\simeq V_{\rm
up}\gg\Lambda$,

$\begin{split}m_{\rm KK}^{\rm bulk}<\frac{3}{p}m_{\rm
Pl}^{1/3}m_{3/2}^{2/3}.\end{split}$ (41)

On the other hand, when $m_{\rm sc}=m_{s}$, the inequality (28) is written as

$\begin{split}3m_{\rm Pl}^{2}m_{3/2}^{2}>|V_{\rm AdS}|>\frac{4\pi
p}{g_{s}}m_{s}^{4}-\frac{3}{s(s-1)}m_{\rm Pl}^{2}m_{0}^{2},\end{split}$ (42)

and (29) reads

$\begin{split}m_{s}<\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/4}m_{\rm Pl}^{1/2}m_{0}^{1/2}.\end{split}$ (43)

Putting an explicit expression for $m_{0}$ given by (27) into the inequality,
we obtain the bound on the strongest warp factor :

$\begin{split}\frac{m_{s}}{m_{\rm Pl}}<\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/4}\Big{(}\frac{m_{0}}{m_{\rm
pl}}\Big{)}^{1/2}=\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/4}\Big{(}\frac{|z_{\ell}|^{1/6}}{M_{\ell}^{1/2}{\cal
V}^{1/6}}\Big{)}.\end{split}$ (44)

For $|V_{\rm AdS}|\simeq V_{\rm up}\gg\Lambda$, we have a more stringent bound
on $m_{s}$,

$\begin{split}m_{s}<\Big{(}\frac{3g_{s}}{4\pi p}\Big{)}^{1/4}m_{\rm
Pl}^{1/2}m_{3/2}^{1/2}.\end{split}$ (45)

We can also compare our lower bound on $m_{s}$ given by (43) with the lower
bound on $m_{s}$ considered in [52, 53]. This comes from the observation that
the mass and spin of string excitations satisfy the Regge trajectory relation,

$\begin{split}m^{2}=(s-1)m_{s}^{2},\end{split}$ (46)

or $m^{2}\simeq sm_{s}^{2}$ for large $s$. This relation, however, violates
the Higuchi bound $m^{2}>s(s-1)H^{2}\simeq s^{2}H^{2}$ when the spin is larger
than $s_{\rm max}=(m_{s}/H)^{2}$, which implies that the cutoff scale is lower
than $\sqrt{s_{\rm max}}m_{s}=m_{s}^{2}/H$. If we identify the cutoff scale
with $m_{\rm Pl}$, we obtain the inequality $m_{s}>H^{1/2}m_{\rm Pl}^{1/2}$.
Combining this with (44), we obtain the bound

$\begin{split}\frac{H}{m_{\rm Pl}}<\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/2}\frac{m_{0}}{m_{\rm
Pl}}=\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/2}\Big{(}\frac{|z_{\ell}|^{1/3}}{M_{\ell}{\cal
V}^{1/3}}\Big{)},\end{split}$ (47)

or roughly, $H<m_{0}$, which is more or less equivalent to the Higuchi bound.
The similar conclusion can be drawn by combining

$\begin{split}3m_{\rm Pl}^{2}m_{0}^{2}>3m_{\rm Pl}^{2}m_{3/2}^{2}>|V_{\rm
AdS}|=\frac{4\pi p}{g_{s}}m_{s}^{4}-3m_{\rm Pl}^{2}H^{2}\end{split}$ (48)

with $m_{s}>H^{1/2}m_{\rm Pl}^{1/2}$ :

$\begin{split}3m_{\rm Pl}^{2}m_{0}^{2}>\Big{(}\frac{4\pi
p}{g_{s}}-3\Big{)}m_{\rm Pl}^{2}H^{2}.\end{split}$ (49)

Meanwhile, if the cutoff scale is given by the species scale $\Lambda_{\rm
sp}\simeq m_{\rm Pl}^{2/3}m_{0}^{1/3}$, we obtain $m_{s}>H^{1/2}\Lambda_{\rm
sp}^{1/2}>H^{1/2}m_{\rm Pl}^{1/3}m_{0}^{1/6}$. Combining this with (44) gives

$\begin{split}\frac{Hm_{0}^{1/3}}{m_{\rm
Pl}^{4/3}}<\Big{(}\frac{3g_{s}(s^{2}-s+1)}{(4\pi
p)s(s-1)}\Big{)}^{1/2}\Big{(}\frac{m_{0}}{m_{\rm pl}}\Big{)},\end{split}$ (50)

or roughly, $m_{0}>(H/m_{\rm Pl})^{3/2}m_{\rm Pl}$, while with (48) gives the
trivial bound

$\begin{split}3m_{\rm Pl}^{2}m_{0}^{2}>3m_{\rm Pl}^{2}m_{3/2}^{2}>\frac{4\pi
p}{g_{s}}m_{s}^{4}-3m_{\rm Pl}^{2}H^{2}>\frac{4\pi p}{g_{s}}H^{2}m_{\rm
Pl}^{4/3}m_{0}^{2/3}-3m_{\rm Pl}^{2}H^{2},\end{split}$ (51)

since the rightmost term is negative.

We close this section by pointing out that the spin $s$ in the Regge
trajectory relation (46) is identified with the level of string excitations in
the Minkowski background. Moreover, the massive field in Minkowski space is
obtained by the Poincaré contraction (taking $H\to 0$ limit) of the
representation in the principal series in which the squared dS mass given by
$\big{(}s-\frac{1}{2}\big{)}^{2}H^{2}+\nu^{2}H^{2}$. Since $\nu/H$ is the mass
of the field in Minkowski space, one may be tempted to identify $\nu/H$,
rather than the dS mass, with the mass in (46), $\sqrt{s-1}m_{s}$. In this
case, the additional term $(s-\frac{1}{2})^{2}H^{2}$ in the squared dS mass is
regarded as the effect of interaction with the background geometry. Then the
condition $sm_{s}^{2}>s^{2}H^{2}$ can be interpreted as follows. So far as the
model for dS space based on the four-dimensional particle description is
concerned, $m_{s}$ as well as $m_{\rm KK}$ is larger than $H$. That is, for
the string excitations, $\nu^{2}H^{2}\simeq sm_{s}^{2}$ in the squared dS mass
is typically dominant over $(s-\frac{1}{2})^{2}H^{2}$ such that if $H\ll
m_{s}$, the dS mass is approximated by the mass in the Minkowski background.
This approximation breaks down when $s>s_{\rm max}\simeq(m_{s}/H)^{2}$, i.e.,
the condition $sm_{s}^{2}>s^{2}H^{2}$ is violated. In this case, the dS mass
can be approximated by $(s-\frac{1}{2})H$ implying that neglecting $H$ is no
longer a good approximation even if $H\ll m_{s}$.

##  4 Conclusions

In this article, we investigate the particular case of Type IIB orientifold
compactification with fluxes in which the internal manifold contains a number
of throats and the warping of the throat containing $\overline{\rm D3}$-branes
is not the strongest. Then the tower mass scale $m_{\rm sc}$ satisfying the
scaling law with respect to the uplift potential generated by $\overline{\rm
D3}$-branes is not the lowest, but has the upper bound determined by the
lowest tower mass scale $m_{0}$, typically given by the KK mass scale
associated with the throat of the strongest warping. This also may be
interpreted as the lower bound on $m_{0}$ determined by $m_{\rm sc}$. When the
exponent $\alpha$ in the scaling law $m_{\rm sc}\sim V_{\rm up}^{\alpha}$ is
$1/4$, the inverse of the number of noncompact spacetime dimensions over which
$\overline{\rm D3}$-branes are extended, the upper bound on $m_{\rm sc}$ is
given by $\sim m_{\rm Pl}^{1/2}m_{0}^{1/2}$. This shows that if $m_{0}$ is
about $10$ TeV, just above the scale accessible at the LHC search, $m_{\rm
sc}$ cannot be higher than the intermediated scale, $\sim 10^{11}$ GeV. This
bound is applied to $m_{\rm sc}=m_{\rm KK}^{\rm throat}$ when the throat
containing $\overline{\rm D3}$-branes is strongly warped and $m_{\rm
sc}=m_{s}$ when the throat containing $\overline{\rm D3}$-branes is extremely
weakly warped. On the other hand, when the throat containing $\overline{\rm
D3}$-branes is extremely weakly warped, $\alpha=1/3$ is allowed for $m_{\rm
sc}=m_{\rm KK}^{\rm bulk}$. In this case, the upper bound on $m_{\rm sc}$ is
given by $m_{\rm Pl}^{1/3}m_{0}^{2/3}$ which is about $5\times 10^{8}$ GeV
when $m_{0}\simeq 10$ TeV. We also point out that the cosmological constant in
our universe can be much smaller than the uplift potential, which allows the
stronger upper bound on $m_{\rm sc}$ in which $m_{0}$ is replaced by the lower
scale, $m_{3/2}$. These bounds tell us how the structure of the internal
manifold is reflected in the relations between different tower mass scales. In
particular, our setup in which the internal manifold contains a number of
throats and the warping of the throat associated with uplift is not the
strongest predicts that the evidences of the extra dimensions as well as the
string may be found under the intermediate scale depending on the value of
$m_{0}$.

### Acknowledgements

This work is motivated by Eran Palti’s question about the author’s parallel
session talk at SUSY 2023 conference.

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