###### Abstract

We study properties of moduli stabilization in the four dimensional supergravity theory with heavy moduli and would-be saxion-axion multiplets including light string-theoretic axions. We give general formulation for the scenario that heavy moduli and saxions are stabilized while axions remain light, assuming that moduli are stabilized near the supersymmetric solution. One can find stable vacuum, i.e. non-tachyonic saxions, in the non-supersymmetric Minkowski vacua. We also discuss the cases, where the moduli are coupled to the supersymmetry breaking sector and/or moduli have contributions to supersymmetry breaking. Futhermore we study the models with axions originating from matter-like fields. Our analysis on moduli stabilization is applicable even if there are not light axion multiplets.

DESY 11-094

KUNS-2340

February 23, 2021

Note on moduli stabilization,

supersymmetry breaking and axiverse

Tetsutaro Higaki^{1}^{1}1
E-mail address: and Tatsuo Kobayashi^{2}^{2}2
E-mail address:

DESY Theory Group, Notkestrasse 85, D-22607 Hamburg, Germany

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

## 1 Introduction

Moduli stabilization in superstring theories compactified on the internal space is necessary to determine physical parameters such as gauge couplings [1], Yukawa couplings [2, 3] and soft supersymmetry (SUSY) breaking parameters [4] in the visible sector, and to evade the moduli problem [5] and undesirable new forces [6]. As a consequence, it also can give several interesting implications to particle physics [7, 8, 9, 10, 11, 12], through the KKLT proposal [13] or the racetrack model [14].

The complex moduli fields in four dimension typically consist of scalars originating from
geometry of compactification space (e.g. its volume) and pseudo-scalars coming from NSNS or RR tensor fields.
Even though all the scalars are stabilized,
some of their partners can still remain light due to the
shift symmetries: const.
Therefore the latter pseudo-scalars are often called string-theoretic axions
[15, 16, 17]
and can include the QCD axion to solve the strong CP problem [18, 19, 20]^{1}^{1}1
If we are to identify one of the axions with the QCD axion,
the quality of the PQ symmetry needs to be checked for solving the strong CP problem:
.
Here axion mass is a contribution from non-QCD effects,
is the QCD axion mass just from the instanton,
is the decay constant of the QCD axion and MeV is the QCD scale.
.
The number of these axions are originally determined by the topological property of compactified space,
e.g. the Hodge numbers of Calabi-Yau (CY) three-fold
[21].
(See also for effective field theories [22, 23].)
Because the numbers can be much larger than of order unity,
one can find many light string-theoretic axions through the moduli stabilization, that is, the string axiverse [24].
The axions can have large axion decay constants beyond the axion window [25]^{2}^{2}2
In the LARGE volume scenario [9],
one can find GeV [16].
and can give influences
on the cosmological observations [24].
For instance, their misalignment angles and Hubble scale during inflationary epoch are constrained and future observations
of tensor modes and isocurvature perturbations
could suggest the evidence of the (non-)axiverse [26].
Of course, the relic abundance of the axions should not
exceed the observed matter density [27].
This will give interesting constraint not only on the observations but also on the string models in terms of moduli stabilization.
Therefore our purpose is to study general framework
of moduli stabilization leading to light axions
based on the supergravity (SUGRA).

Besides string-theoretic axions, one often obtains light field-theoretic axions at low energy, too. Thus, in general, the number of axions is estimated as [28]

(the number of axions) | (the number of fields) | |||

Here is the superpotential. This is because the Peccei-Quinn (PQ) shift symmetries of fields and the -symmetry produce candidates of the axions whereas independent terms in the superpotential kill them, assuming the Kähler potential preserves these symmetries. Even if the -symmetry is broken explicitly, this estimate is consistent when the constant in the superpotential is involved in the term ”the number of terms in the ”. Although we have neglected vector multiplets which can become massive, they can also reduce the number of axion candidates by absorbing them. When this counting becomes negative or zero, we do not have any light axions. If there are very small terms violating PQ symmetries in or , they give very light masses to the axions.

In this paper, we study the moduli stabilization scenario leading to light axions. We discuss conditions to give heavy masses to all of real parts of moduli and leave some of imaginary parts massless. One of important conditions is SUSY breaking, and the typical mass scale is the gravitino mass . All of the real parts of moduli must have masses, which are larger than the gravitino mass and/or comparable to the gravitino mass. On the other hand, light axions masses are smaller and could be of with or a few tens.

In Section 2, we will study the properties of non-supersymmetric vacua with light string-theoretic axions. We will also give comments on closed string moduli which are directly coupled to the SUSY breaking sector. In Section 3, we will study the string-theoretic -axion and the saxion-axion multiplet breaking SUSY. In Section 4, we will discuss corrections to the light axion masses from small breaking terms of PQ symmetries in the superpotential and the Kähler potential. In Section 5, we will give comments on simple models of field-theoretic axions in terms of effective field theories. In Section 6, we will conclude this paper. Our analysis on moduli stabilization is applicable even if there are not light axion multiplets. In Appendix, several types of moduli stabilization models are briefly reviewed. We will give a brief comment on the LARGE volume scenario based on the recent work of the neutral instanton effect including odd parity moduli under orientifold parity.

## 2 Light string-theoretic axions

In the following sections, we will consider moduli stabilization at low energy with the assumption that irrelevant moduli are heavy by closed string fluxes [29]. The remaining moduli of our interest can be stabilized via gaugino condensation [30] or (stringy) instanton effects [31]. Thus we study the superpotential below:

(2.1) |

Here is a constant from the fluxes, are heavy closed string moduli fields which are stabilized by this superpotential and we use the unit GeV . We study the possibility that we can have massless axions at this stage. The scalar potential is written by the superpotential and the Kähler potential ,

where

(2.3) |

Here, denotes the inverse of the Kähler metric . F-terms and the gravitino mass are given as

(2.4) |

We will focus just on for simplicity.

### 2.1 Light string-theoretic axions and saxion masses in the SUSY vacuum

In this subsection, we briefly review [16]. We study saxion masses in the SUSY vacuum with light axions.

For instance, let us consider the superpotential with two moduli :

(2.5) |

One can find is absent from the superpotential, that is, we have just one phase of : . Then the imaginary part Im is a massless axion whereas Re may be stabilized via the Kähler potential .

One can generalize this argument to the case with many axions. Chiral superfields are classified into two classes. One class of fields do not appear in the superpotential, i.e.

(2.6) |

while the fields in the other class appear. Then, the imaginary parts of , i.e. are string-theoretic axions, which have flat directions in the scalar potential for the form of Kähler potential, . We evaluate masses of the real parts of , i.e. saxions . In the SUSY vacuum with stabilized moduli one finds

(2.7) |

For the fields , this leads to

(2.8) |

In this case, we find

(2.9) |

That is, every massless string-theoretic axion has undesirable massless saxion for or tachyonic saxion in the SUSY AdS vacuum for . This is because is the positive definite matrix. Note that the term comes from the vacuum energy. We have used the property of perturbative moduli Kähler potential,

(2.10) |

The tachyonic instability might not be problematic in the AdS vacuum
because of the Breitenlohner-Freedman bound
[32].
At any rate, one should consider the SUSY breaking Minkowski vacuum
to realize the realistic vacuum,
although one may need fine-tuning
to uplift the SUSY AdS vacuum to the Minkowski one.
Hence, in the following sections,
we will consider the SUSY breaking effects and then
one can see that the saxions become stable for
vanishing vacuum energy ^{3}^{3}3
One can also consider a non-perturbative effect on the Kähler potential or
D-term moduli stabilization which means a gauge multiplet eats an axion multiplet to lift saxion direction.
.

### 2.2 Light string-theoretic axions and the saxion mass in the SUSY breaking Minkowski vacuum

Here, we study saxion stabilizaton in the SUSY breaking Minkowski vacuum with light axions. As a SUSY breaking source, we consider a single chiral field . We assume that moduli -terms are smaller than and the cosmological constant is vanishing, , that is,

(2.11) |

where .

Here, we study the model, where the SUSY breaking sector and moduli are decoupled in the Kähler potential and the superpotential . That is, we consider the following form of the Kähler potential and the superpotential

(2.12) |

Hereafter we will set at the leading order of
.
Note that and
.
When there is a large mass splitting between moduli and ,
would be possible,
but would be necessary for the stable vacuum;
would be an appropriate approximation.
A simple example of the SUSY breaking models has
[34, 35, 36, 37]^{4}^{4}4
There are also models including SUSY breaking moduli
[38], but we will not consider such models
since subtle fine-tuning would be necessary.
.
At any rate, here we consider generic form of the SUSY breaking
superpotential .

From the above assumption, one expects moduli and are stabilized near the SUSY solution,

(2.13) |

such that one obtains heavier moduli masses than the gravitino mass . In the SUSY breaking vacuum with a vanishing cosmological constant, one finds the stationary condition:

(2.14) |

which leads to

(2.15) |

Here denotes , and is a covariant derivative with respect to the Kähler metric. Since , the above equation becomes

(2.16) |

Here, we have used

(2.17) |

because and eq. (2.11). Using , one finds in the vacuum

(2.18) |

This means -term of is suppressed unless there is mixing between and . For and , one can typically neglect sub-leading terms

(2.19) |

and one obtains

(2.20) |

Here is an inverse matrix of . Thus one can expect the shifts from the SUSY solution of and are given by

(2.21) |

Here we have used typical results and . One will see these shifts can be suppressed by the heavy moduli masses squared as .

#### 2.2.1 Masses for sGoldstino and heavy moduli

We evaluate masses of and . By differentiating eq.(2.14), we obtain in the vacuum

(2.22) |

where

(2.23) |

Since we assumed that heavy moduli are stabilized near the SUSY solution, one can neglect term to calculate heavy moduli masses at the leading order of SUSY breaking effect.

For example, one expects

(2.24) |

for the KKLT-like stabilization [13] and

(2.25) |

for the racetrack model [14], which is viable even for . (See also Appendices A.2 and A.4 for the KKLT-like stabilization and the racetrack model, respectively.) Here denotes the most effective (or smallest) one in appearing in the eq.(2.1) to the moduli mass . One could obtain heavier moduli masses than the gravitino mass by fine-tuning the constant in the racetrack model [39].

In general, one expects and mass squared matrix elements of the moduli are written as

(2.26) |

that is,

(2.27) |

for . Note that the mass is the supersymmetric mass of modulus . In the above, we have used the following approximation,

(2.28) |

We took the diagonal mass matrix for simplifying the discussion here. Also one finds

(2.29) |

For with for any , their values are estimated as . Here we have used no-scale like structure up to would-be small perturbative corrections, though there is the small dependencies . Note that the contribution of eq.(2.28) to can be comparable to supersymmetric case, but one still has . Thus, one can obtain the (perturbatively) stable minimum for proper values of the moduli masses, . That is, by making larger than , one can realize positive definite mass eigenvalues for all of moduli around the SUSY solution . Indeed, by using the above result, it is found the shift in (2.21) is suppressed by the factor, .

Next, we evaluate the mass of sGoldstino . The sGoldstino acquires not the mass from but only SUSY breaking mass from the Kähler potential because of massless Goldstino in the rigid limit. There is the necessary condition (not sufficient) for the stable SUSY breaking vacuum, i.e. non-tachyonic non-holomorphic sGoldstino mass [38, 40]:

(2.30) |

where

(2.31) |

For one expects

(2.32) |

For instance, let us consider the Kähler potential with a heavy scale [41, 35, 36]

(2.33) |

Then one obtains

(2.34) |

Here would be of for the Polonyi model. For off-diagonal component , so long as and are of order unity in the Planck unit, one can find . Thus, there would be the stable minimum. For string theories, would correspond to the mass scale of heavy field which is coupled to , such as anomalous gauge multiplet mass [42] which is comparable to the string scale, when has the charge.

#### 2.2.2 Masses for saxion

Here, we evaluate masses of saxion . One finds positive mass squared:

(2.35) |

Here we have neglected the last three terms in the bracket, since when one obtains one can find

(2.36) |

Again we have used no-scale like structure Then the last three terms in eq.(2.35) are suppressed by and respectively, compared to the first term.

Instead of , with the sequestered explicit SUSY breaking term where , one finds the similar results [17], and , i.e. . Here we have neglected the term which is proportional to in . Note also that mass spectra of heavy moduli for such a case are similar to ones discussed above.

#### 2.2.3 Matrix elements

Here, we summarize the mass matrix. Including other matrix elements, one can find typically

(2.37) | |||||

where we have used

(2.38) |

as well as . In general, and could cause the vacuum instability even if and . Based on these matrix elements one expects the conditions

(2.39) |

should be satisfied for the (meta)stability. For this case, so long as one would obtain the stable minimum. Then, the mass spectrum is summarized as

(2.40) |

At this stage, the axions are massless. Note that all of saxions corresponding to massless axions have almost the same mass .

Here, after the Goldstino is absorbed into the gravitino, the unnormalized axino masses are given by

(2.41) | |||||

We have neglected and because they are of corrections.

#### 2.2.4 -term

In the above case, one can find

(2.42) |

Here we used the result

(2.43) |

which leads to . Even if any are stabilized via -terms, , we gain -term of the through the off-diagonal Kähler metric [43, 44, 45]. Note that if , one finds since for such a case [46]. For string-theoretic axion(s) breaking SUSY, see the Section 3.2.

### 2.3 Note on mixing between and moduli and -terms

For simplicity, we have discussed so far the case that the SUSY breaking field does not couple to moduli for a simplicity. However, in string theories, it is natural that moduli are coupled to the SUSY breaking sector via non-perturbative effects, so that one obtains much smaller scale than the string scale. Now, let us consider the mixing between and heavy moduli by replacing in (2.12) as follows,

(2.44) |

Here depends only on . For instance one can consider the case that [36, 37] or [10]. Then, we consider the moduli stabilization with the superpotential,

(2.45) |

We assume

(2.46) |

in the above superpotential, where is the most effective one to the moduli mass in for . Then one can find

(2.47) |

Also, one obtains for

(2.48) | |||||

and also we estimate

(2.49) |

For metastability, one expects the conditions (2.2.3) should be satisfied.

Here, with the assumption that , one finds for