6 August, 2007 at SI2007 .

1. 6 August, 2007 at SI2007. Moduli stabilization, F-term uplifting and soft supersymmetry breaking terms Tetsutaro Higaki (Tohoku University) Based on articles : • Phys.Rev.D75:025019,2007, • ArXiv:0707.2671 [hep-th]. In collaboration with Hiroyuki Abe (YITP), Tatsuo Kobayashi (Kyoto U.) and Yuji Omura (Kyoto U.)

2. : gauge and Yukawa couplings : SUSY breaking soft terms 1. Introduction and Motivation • Superstring theoryhas some good properties: • Candidate of unified theory of matter and forces. • No freecontinuousparameters. Vacuum expectation values of moduli superfields can determine physical parameters. (moduli mediation) Moduli are important for particle phenomenology. In this talk, we set

3. Once moduli are stabilized , we can generally obtain negative cosmological constant. However the realistic value is very tiny butpositive. Therefore we want to uplift the potential by adding uplifting sector. In this talk, we will pay our attention to the overall volume modulus Twhich determines compactification scale (based on type IIB string/SUGRA on CY orientifold) .

4. 2. KKLT model: S.Kachru, R.Kallosh, A.Linde and S.P.Trivedi This simplified model is based on warped compactification. In the low energy scale, we have Bousso et al, K.Choi et al. Gravitino mass Furthermore we have aexplicit SUSY termlocalizedon the intermediate scale brane, which uplifts SUGRA potential : The minimum of the warp factor. Giddings et al. Zero C.C. Stabilized anti-D3-brane (intermediate scale brane) MSSM and strong coupling sector on D7-branes (assumption) Sequestered uplifting K.Choi, A. Falkowski, H.P.NIlles,M.Olechowski

5. Then we find the whole potential : At the SUSY vacuum of Shift from the SUSY vacuum 0 Finely tuned cosmological constant Mass scales S. Weinberg ( 4D cut off scale)

6. SUSY breaking order parameters The modulus mediation iscomparableto the anomaly mediation = mirage mediation K.Choi et al, Endo et al. This mediation mechanism has a interesting feature : Pure modulus mediation appears at the following scale. Modulus mediation AMSB

7. KKLT (α=1) For in MSSM model(1st and 2nd generations) K.Choi arxiv:0705.3330 Then, we find that all moduli can be fixed and we can have distinct sparticle spectrum, though this model has explicit SUSY breaking mechanism. One may feel uneasy.

8. 3. F-term uplifting There are various uplifting schemes, which breaks SUSY spontaneously. ・D-term uplifting : this cannot work simply C.P. Burgess et al. ・Kaehler uplifting A. Westphal,… In the above case of both, we can have ・F-term uplifting(adding SUSY breaking sector X) Lebedev et al., Dudas et al.,Abe et al., Kallosh et al. In this case, we can obtain

9. SUSY breaking models (candidates for F-term uplifting sector) (i) Polonyi model This model can have SUSY breaking Minkowski vacuum:

10. (ii) Quantum corrected (local) O’raifeartaigh model (ex) ・Intriligator-Seiberg-Shih (ISS) model ・Izawa-Yanagida-Intriligator-Thomas model ISS This model have SUSY breaking Minkowski vacuum, too. ISS

11. 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? Dine et al. : Polonyi or ISS like vacuum, : KKLT vacumm.

12. Estimation of F-term of modulus T where we assumed Almost diagonal metric

13. We define Finally we find With a equation of motion of , we obtain From

14. At first, we consider the case with

15. is needed. • Polonyi-KKLT model • The true minimum is expressed by Here we used , (AMSB contribution)

16. is needed. • (ii) ISS-KKLT model • The true minimum is expressed by Here we used , (AMSB contribution)

17. Next, we consider the case with Florea et al., Ibanez et al. Blumenhagen et al. We want to consider • (i) Polonyi-KKLT model • (ii) ISS-racetrack model This term is needed in order to make the reference point valid.

18. 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

19. (Polonyi or ISS)-KKLT (α=2/3) 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

20. 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 Heavy scalar Mirage for ISS case Abel, Blumenhagen In case of both (seq. or not seq.), b-term is expressed by We may need somewhat tuning for EWSB (mirage case). ∃

21. 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

22. In the racetrack type model, we can obtain much largermass of modulus GeV than gravitino mass GeVand one of KKLT case GeV. In spite of such large mass, we can obtain moderate value of F-component of modulus T through instanton effects which depend on T. 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))

23. 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 theoryetc.

24. Appendix

25. ● Gaugino mass at ● SUSY breaking scalar mass at ● A-term (coefficients of 3-point of scalar) at

26. Gaugino masses at TeV scale : K.Choi,K.Jeong,K.Okumura, JHEP09 (2005) 039 K.Choi arxiv:0705.3330 = (1 + 0.66α) : (2 + 0.2α) : (6 − 1.8α)

27. For 1st and 2nd generation with Sfermion masses at TeV scale : K.Choi,K.Jeong,K.Okumura, JHEP09 (2005) 039 K.Choi arxiv:0705.3330 : = : :

28. An evaluation of from the reference point In the previous two examples, we have Real O.Lebedev et al., is determined. can be done always by field redefinition.

29. 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.

30. 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.

31. Therefore, model (i) is racetracktype 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.

32. Example 1

33. Example 2

34. Example 3

35. Example 4

36. Example 5

37. For we have α=O(1). For we have very large, i.e., α >> 1. In this case, wefind that anomaly mediation is dominant, and

38. 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