6 August, 2007 at SI2007.

• Phys.Rev.D75:025019,2007,

• ArXiv:0707.2671 [hep-th].

Based on articles :

In collaboration with Hiroyuki Abe (YITP), Tatsuo Kobayashi (Kyoto U.)

and Yuji Omura (Kyoto U.)

Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms

Tetsutaro Higaki(Tohoku University)

1. Introduction and Motivation

Superstring theory has some good properties:

• Candidate of unified theory of matter and forces.

• No free continuous parameters.Vacuum expectation values of moduli superfields

can determine physical parameters.

( moduli mediation )

Moduli are important for particle phenomenology.

In this talk, we set

: gauge and Yukawa couplings

: SUSY breaking soft terms

Once moduli are stabilized ,we can generally obtain negative cosmological constant.However the realistic value is very tiny but positive.

Therefore we want to uplift the potential by adding uplifting sector.

In this talk, we will pay our attention to the overall volume modulus T which determines compactification scale(based on type IIB string/SUGRA on CY orientifold) .

This simplified model is based on warped compactification.

2. KKLT model: S.Kachru, R.Kallosh, A.Linde and S.P.Trivedi

In the low energy scale, we have

Furthermore we have a explicit SUSY term localized on the intermediate scale brane, which uplifts SUGRA potential

: The minimum of the warp factor.

Stabilized anti-D3-brane (intermediate scale brane)MSSM and

strong coupling sectoron D7-branes(assumption)

Sequestered uplifting

Bousso et al,K.Choi et al.

Giddings et al.

Gravitino mass

Zero C.C.

Then we find the whole potential :

At the SUSY vacuum of

0

( 4D cut off scale )

Mass scales

Shift from the SUSY vacuum

Finely tuned cosmological constantS. Weinberg

The modulus mediation is comparable to the anomaly mediation

= mirage mediation

SUSY breaking order parameters

This mediation mechanism has a interesting feature :Pure modulus mediation appears at the following scale .

K.Choi et al, Endo et al.

AMSB

Modulus mediation

For in MSSM model(1st and 2nd generations)K.Choi arxiv:0705.3330

KKLT (α=1)

Then, we find that all moduli can be fixed and we can have distinctsparticle spectrum, though this model has explicit SUSY breaking mechanism.

One may feel uneasy.

3. F-term uplifting There are various uplifting schemes, which breaks SUSY spontaneously.

・ D-term uplifting : this cannot work simply

・ Kaehler uplifting

In the above case of both, we can have

C.P. Burgess et al.

A. Westphal,…

・ F-term uplifting (adding SUSY breaking sector X)

In this case, we can obtain

Lebedev et al., Dudas et al.,Abe et al., Kallosh et al.

(i) Polonyi model

This model can have SUSY breaking Minkowski vacuum :

SUSY breaking models (candidates for F-term uplifting sector)

(ii) Quantum corrected (local) O’raifeartaigh model (ex) ・ Intriligator-Seiberg-Shih (ISS) model ・ Izawa-Yanagida-Intriligator-Thomas model

This model have SUSY breaking Minkowski vacuum, too.

ISS

ISS

Moduli stabilization and F-term uplifting

We study a combination of the previous two types of SUSY breaking model and KKLT type moduli stabilization, that is,

(i) Polonyi-KKLT model :

or

(ii) ISS-KKLT model :

Now the scalar potential is somewhat complicated, so we examine it around the reference point such that

And how about mirage mediation?

: Polonyi or ISS like vacuum,: KKLT vacumm.

Dine et al.

Estimation of F-term of modulus T

where we assumed

Almost diagonal metric

We define

Finally we find

With a equation of motion of , we obtain

From

At first, we consider the case with

(i) Polonyi-KKLT model

The true minimum is expressed by

Here we used ,

(AMSB contribution)

is needed.

(ii) ISS-KKLT model

The true minimum is expressed by

Here we used ,

(AMSB contribution)

is needed.

Next, we consider the case with

(i) Polonyi-KKLT model

(ii) ISS-racetrack model

Florea et al., Ibanez et al. Blumenhagen et al.

We want to consider

This term is needed in order to make the reference point valid.

Because superpotential of modulus is now racetrack type,we have

Without an enhancement factor b, we find However with the factor we can get typically

Then we can typically find α=O(1) (in paper).(In ISS-racetrack model of some cases, we can obtain α>>1. )

Kallosh, Linde

4. Soft SUSY breaking terms

When X field is sequestered from SM fields, we can obtain particle spectra of the mirage mediation scenario with

For the case with

For in MSSM model(1st and 2nd generations)K.Choi arxiv:0705.3330

(Polonyi or ISS)-KKLT (α=2/3)

When X field is not sequestered, we may obtain soft mass from gravity mediation sector

Here we supposed that Yukawa coupling is constant for T.For 1st and 2nd generation, if Yukawa coupling is given by

we obtain larger A-term than one of 3rd generation

In case of both (seq. or not seq.), b-term is expressed by

We may need somewhat tuning for EWSB (mirage case).

Abel, Blumenhagen

Mirage for ISS case

∃

Heavy scalar

SUSY CP phase :

But ISS-racetrack model with constant is not so.

Gauge mediation mechanism can be dominant, when there are proper couplings between X field and messenger fields which are charged under the SM gauge group. (The value of some parameters should be changed.) Kitano (talk), S.P.de Alwis

5.Conclusion

We performed KKLT type modulus stabilization with F-term uplifting scheme (spontaneous SUSY breaking) instead of explicit SUSY breaking term.

Low energy spectra of SUSY particles can depend on the couplings of X to SM sector. (Mirage or heavy scalar or gauge mediation (parameters changed from gravity mediation case))

In the racetrack type model, we can obtain much larger massof modulus GeV than gravitino mass GeV and oneof KKLT case GeV. In spite of such large mass, we can obtain moderate value of F-component of modulus T throughinstanton effects which depend on T.

6. Open question and problem

・ Sequestering of SM sector from X・ Study for gravitino overproduction problem (Polonyi-KKLT with constant μ is already studied by Dine et al.)・ Concrete realization of strong couplig sector (computation of instanton effects) in string theory・ Concrete realization of F-term uplifting sector X in string theory (is stabilization of another open string moduli needed?)・ Concrete realization of SM sector in string theory 　 etc.

Appendix

● Gaugino mass at

● SUSY breaking scalar mass at

● A-term (coefficients of 3-point of scalar) at

Gaugino masses at TeV scale :K.Choi,K.Jeong,K.Okumura, JHEP09 (2005) 039

= (1 + 0.66α) : (2 + 0.2α) : (6 − 1.8α)

K.Choi arxiv:0705.3330

Sfermion masses at TeV scale :K.Choi,K.Jeong,K.Okumura, JHEP09 (2005) 039

:: :

=

For 1st and 2nd generation with K.Choi arxiv:0705.3330

An evaluation of from the reference point

In the previous two examples, we have

O.Lebedev et al.,

is determined.

Real

can be done always by field redefinition.

Remark

(i)

In this case, we have

Then the reference point is unstable. The true vacuum is far from the Polonyi vacuum. (Expansion from the reference point cannot converge.)

The case of (ii) is stable for such deformation.

An evaluation of from the reference point

Here let’s think the following superpotential for just modulus

At the SUSY vacuum, we find

We have larger hierarchy between and rather than KKLT type superpotential in this racetrack type one.

Therefore, model (i) is racetrack type modulus stabilization, while model (ii) is KKLT type modulus stabilization due to the smallness of X at the reference point. This results in the fact that the reference point in model (ii) is unstable (expansion from the reference point cannot converge).

We should think ISS-racetrack model instead of ISS-KKLT model.

Example 1

Example 2

Example 3

Example 4

Example 5

For we have α=O(1).

For we have very large, i.e., α >> 1. In this case,we find that anomaly mediation is dominant,

and

This is the case with a superpotential:

The model with this superpotential has the (SUSY AdS ) vacuum:

Even after one uplifts potential, we have much smaller gravitino mass than mass of modulus.

Kallosh, Linde