Moduli stabilization, SUSY breaking and the Higgs sector Tatsuo Kobayashi １． Introduction 2. KKLT scenario 3 ． Generalized KKLT scenario 4. TeV scale mirage mediation 5. Summary based on based on Choi, Jeong, T.K., Okumura, hep-ph/ , Choi, Jeong, T.K., Okumura, hep-ph/ , Abe, Higaki, T.K., hep-th/ , , Abe, Higaki, T.K., hep-th/ , , Abe, Higaki, T.K., Omura, hep-th/ Abe, Higaki, T.K., Omura, hep-th/ Abe, T.K., Omura, hep-ph/ Abe, T.K., Omura, hep-ph/ Abe, Kim, T.K., Shimizu, [hep-ph] Abe, Kim, T.K., Shimizu, [hep-ph]

1. Introduction Motivation for low-energy SUSY the hierarchy problem, naturalness problem the hierarchy problem, naturalness problem gauge coupling unification gauge coupling unification dark matter dark matter SUSY should be broken and SUSY should be broken and superpartners should have larger masses. superpartners should have larger masses. Phenomenological aspects depend on Phenomenological aspects depend on a pattern of SUSY breaking terms, a pattern of SUSY breaking terms, gaugino masses, sfermion masses, etc. gaugino masses, sfermion masses, etc. Such a pattern is determined by how SUSY Such a pattern is determined by how SUSY breaking is mediated to the visible sector. breaking is mediated to the visible sector.

SUSY breaking mediation Several mediation mechanisms of SUSY breaking Supergravity mediation (mSUGRA) Supergravity mediation (mSUGRA) (moduli-dilaton mediation) (moduli-dilaton mediation) Gauge mediation Gauge mediation Anomaly mediation Anomaly mediation New type of mechanism New type of mechanism Mirage mediation Mirage mediation This talk covers theoretical side and This talk covers theoretical side and discusses some phenomenological aspects. discusses some phenomenological aspects. Mirage mediation can be realized in Mirage mediation can be realized in string-derived supergravity. string-derived supergravity. TeV scale mirage is quite interesting, and TeV scale mirage is quite interesting, and leads to a unique pattern of s-spectrum. leads to a unique pattern of s-spectrum.

String phenomenology Superstring theory is a promising candidate for unified theory including gravity. for unified theory including gravity. Superstring theory has several moduli fields including the dilaton. including the dilaton. Moduli correspond to the size and shape of compact space. of compact space. VEVs of moduli fields  couplings in low-energy effective theory,  couplings in low-energy effective theory, e.g. gauge and Yukawa couplings e.g. gauge and Yukawa couplings Thus, it is important to stabilize moduli VEVs Thus, it is important to stabilize moduli VEVs at realistic values from the viewpoint of at realistic values from the viewpoint of particle physics as well as cosmology particle physics as well as cosmology Actually, lots of works have been done so far.

Moduli stabilization and SUSY breaking Low-energy effective theory = supergravity Moduli-stabilizing potential may break SUSY. When When Moduli-mediated SUSY breaking is dominant unless couplings between the visible sector and moduli fields are suppressed between the visible sector and moduli fields are suppressed by some reason. by some reason. This is the conventional approach. This is the conventional approach. Gaugino masses and sfermion masses are comparable with the gravitino mass. with the gravitino mass. (Ratios are fixed by supergravity Lagrangian.) (Ratios are fixed by supergravity Lagrangian.) For example, when gauge kinetic functions and Kahler metric are universal, that leads to the spectrum of mSUGRA. metric are universal, that leads to the spectrum of mSUGRA.

Moduli stabilization and SUSY breaking The magnitude of moduli F-components depend on moduli stabilization mechanism. moduli stabilization mechanism. When F/M is smaller than the gravitino mass, another contribution is also important or rather dominant. another contribution is also important or rather dominant. ⇒ Different spectrum of s-particles

KKLT scenario Kachru, Kallosh, Linde, Trivedi, ‘ 03 Kachru, Kallosh, Linde, Trivedi, ‘ 03 They have proposed a new scenario for moduli stabilization leading to de Sitter (or Minkowski) vacua, leading to de Sitter (or Minkowski) vacua, where all of moduli are stabilized. where all of moduli are stabilized. Soft SUSY breaking terms Choi, Falkowski, Nilles, Olechowski, Pokorski ’ 04, CFNO ‘ 05 Choi, Falkowski, Nilles, Olechowski, Pokorski ’ 04, CFNO ‘ 05  a unique patter of soft SUSY breaking terms  a unique patter of soft SUSY breaking terms Modulus med. and anomaly med. are comparable. Modulus med. and anomaly med. are comparable.  Mirage (unification) scale  Mirage (unification) scale Mirage Mediation Choi, Jeong, Okumura, ‘ 05 Mirage Mediation Choi, Jeong, Okumura, ‘ 05  little SUSY hierarchy (TeV scale mirage mediation)  little SUSY hierarchy (TeV scale mirage mediation) Choi, Jeong, T.K., Okumura, ’ 05, ‘ 06 Choi, Jeong, T.K., Okumura, ’ 05, ‘ 06

More about moduli stabilization a generic KKLT scenario with moduli-mixing superpotential with moduli-mixing superpotential ⇒ various mirage scale ⇒ various mirage scale Abe, Higaki, T.K., ’ 05 Abe, Higaki, T.K., ’ 05 Choi, Jeong, ’ 06 Choi, Jeong, ’ 06 Choi, Jeong, T.K., Okumura, ‘ 06 Choi, Jeong, T.K., Okumura, ‘ 06 F-term uplifting Dudas, Papineau, Pokorski, ’ 06 Dudas, Papineau, Pokorski, ’ 06 Abe, Higaki, T.K., Omura, ’ 06 Abe, Higaki, T.K., Omura, ’ 06 Kallosh, Linde, ’ 06 Kallosh, Linde, ’ 06 Abe, Higaki, T.K, ‘ 07 Abe, Higaki, T.K, ‘ 07

2. KKLT scenario This scenario consists of three steps in type IIB. 1) Flux compactification Giddings, Kachru, Polchinski, ‘ 01 Giddings, Kachru, Polchinski, ‘ 01 The dilaton S and complex structure moduli U are assumed to be stabilized by assumed to be stabilized by the flux-induced superpotential the flux-induced superpotential while the Kaher moduli T remain not stabilized. while the Kaher moduli T remain not stabilized. (It is stabilized in the next step.) (It is stabilized in the next step.)

2) Non-perturbative effect We add T-dependent superpotential induced by e.g. gaugino condensation on D7. e.g. gaugino condensation on D7. Scalar potential T is stabilized at D T W=0  SUSY Anti de Sitter vacuum V < 0 Unit : Unit :

Stabilization of T gravitino mass gravitino mass Scalar potential T is stabilized at D T W=0  SUSY Anti de Sitter vacuum V < 0 Moduli mass

3) Uplifting We add uplifting potential generated by e.g. anti-D3 brane at the tip of warp throat e.g. anti-D3 brane at the tip of warp throat The value of D can be suppressed by the warp factor. We fine-tune such that V F +V L = 0 (or slightly positive) V F +V L = 0 (or slightly positive)  SUSY breaking de Sitter/Minkowski vacuum  SUSY breaking de Sitter/Minkowski vacuum T is shifted slightly from the point D T W=0

Configuration in compact space hidden sector hidden sector non-perturbative effect non-perturbative effect T anti D3 anti D3 (SUSY breaking source) (SUSY breaking source) visible sector visible sector

Uplifting V SUSY breaking SUSY breaking total potential total potential T before uplifting before uplifting SUSY point SUSY point

Mirage mediation If aT = O(1), moduli-mediation is dominant However, when aT=O(10), moduli mediation and anomaly mediation are comparable, that is, anomaly mediation are comparable, that is, Mirage mediation. Mirage mediation. Choi, Jeong, Okumura, ‘ 05 Choi, Jeong, Okumura, ‘ 05

Mirage mediation = (modulus med.) + (Anomaly med.) = (modulus med.) + (Anomaly med.) the mirage scale the mirage scale

Mirage mediation For scalar masses, we can obtain the same behavior. behavior. Scalar masses (Mx) = mixture of Scalar masses (Mx) = mixture of (modulus med.) and (Anomaly med.). However, at the mirage scale, However, at the mirage scale, RG effect and Anomaly med. are cancelled each other. RG effect and Anomaly med. are cancelled each other.

Scalar mass due to modulus med. Kahler metric Scalar mass due to modulus mediation

Soft SUSY breaking Modulus med. and anomaly med. are comparable. It is useful to define the ratio, AM/modulus med. An interesting aspect is the mirage scale, where anomaly med. at the cut-off scale and RG effects between the cut-off scale and the mirage scale cancel each other. Original KKLT  α=1 Mirage scale = intermediate scale Mirage scale = intermediate scale α=2 ⇒ TeV scale mirage

example example Suppose that the modulus mediation leads to the s-spectrum of mSUGRA. the s-spectrum of mSUGRA. α=0 ⇒ usual mSUGRA α=0 ⇒ usual mSUGRA α=2 ⇒ TeV scale mirage α=2 ⇒ TeV scale mirage

3. Generalized KKLT 3.1 questions and generalization Each step has further question and generalization. Each step has further question and generalization. 1) Are S and U stabilized really ? 1) Are S and U stabilized really ? What happens if they are not stabilized ? What happens if they are not stabilized ? Is two-step stabilization reliable ? Is two-step stabilization reliable ? If S and U have SUSY masses much larger than the gravitino mass, S and U are stabilized. than the gravitino mass, S and U are stabilized. Abe, Higaki, T.K., ‘ 06 Abe, Higaki, T.K., ‘ 06

2) Moduli mixing superpotential In several string models, gauge kinetic function f is obtained as a linear combination of two or more obtained as a linear combination of two or more fields. fields. Weakly coupled hetero. /heterotic M Similarly, IIA intersecting D-branes/IIB magnetized D-branes IIA intersecting D-branes/IIB magnetized D-branes Lust, et. al. ’ 04 Lust, et. al. ’ 04 Gaugino condensation  exp[-a f] Moduli mixing superpotential What happens with moduli-mixing superpotential ?

3) F-term uplifting Can anti-D3 brane (explicit breaking) be replaced by spontaneous SUSY breaking sector ? by spontaneous SUSY breaking sector ? F-term uplifting F-term uplifting Gomez-Reino, Scrucca, ’ 06 Gomez-Reino, Scrucca, ’ 06 Lebedev, Nilles, Ratz, ’ 06, Lebedev, Nilles, Ratz, ’ 06, Dudas, Papineau, Pokorski, ’ 06 Dudas, Papineau, Pokorski, ’ 06 Abe, Higaki, T.K., Omura, ’ 06 Abe, Higaki, T.K., Omura, ’ 06 Kallosh, Linde, ’ 06 Kallosh, Linde, ’ 06 Abe, Higaki, T.K., ‘ 07 Abe, Higaki, T.K., ‘ 07 Yes, that is possible in a simple model.

3-2. moduli-mixing superpotential In several string models, gauge kinetic function f is obtained as a linear combination of two or more obtained as a linear combination of two or more fields. fields. IIA intersecting D-branes/IIB magnetized D-branes IIA intersecting D-branes/IIB magnetized D-branes Lust, et. al. ’ 04 Lust, et. al. ’ 04 Gaugino condensation  W = exp[-a f] Moduli mixing superpotential S is assumed to be stabilized already.

Moduli mixing Abe, Higaki, T.K., ’ 05 Abe, Higaki, T.K., ’ 05 Choi, Jeong, ‘ 06 Choi, Jeong, ‘ 06 Moduli mixing superpotential ⇒ change F-term of T after uplifting ⇒ change F-term of T after uplifting Gauge kinetic function of the visible sector ⇒ change ratio of gaugino mass to F-term ⇒ change ratio of gaugino mass to F-term Both change a value of α

Our model Choi, Jeong, T.K. Okumura, ‘ 06 Choi, Jeong, T.K. Okumura, ‘ 06 m,n,k,p are discrete. m,n,k,p are discrete. S is replace by its VEV. We also add the same uplifting potential.

Generic results T is stabilized at the AdS SUSY point before uplifting. before uplifting. After adding the uplifting potential, the minimum shifts slightly and the F-term of T the minimum shifts slightly and the F-term of T becomes non-vanishing. becomes non-vanishing. The size of F-term depends on m and n. That leads to a rich structure of SUSY breaking terms, e.g. different patterns of s-particle masses.

Cosmological implications Moduli fields are also important from the viewpoint of cosmology, e.g. inflation. the viewpoint of cosmology, e.g. inflation. Some moduli fields may be a candidate for inflaton. inflaton. Moduli-mixing superpotential has interesting aspects for cosmology. aspects for cosmology.

Cosmology Original KKLT Height of bump is determined by determined by gravitino mass gravitino mass Overshooting problem Brustein, Steinhardt, ‘ 93 Overshooting problem Brustein, Steinhardt, ‘ 93 Inflation ? Inflation ? destabilization due to finite temperature effects destabilization due to finite temperature effects Buchmuller, et. al. ‘ 04 Buchmuller, et. al. ‘ 04

Moduli mixing Abe, Higaki, T.K., ‘ 05 Racetrack model The above problems may The above problems may be avoided. be avoided.

3.2 F-term uplifting Dudas, et. al. ’ 06, Dudas, et. al. ’ 06, Abe, Higaki, T.K., Omura ’ 06 Abe, Higaki, T.K., Omura ’ 06 Kallosh, Linde, ‘ 06 Kallosh, Linde, ‘ 06 Abe, Higaki, T.K., ‘ 07 Abe, Higaki, T.K., ‘ 07 Can we realize uplifting by a spontaneous SUSY breaking ? Can we realize uplifting by a spontaneous SUSY breaking ? If the potential minimum corresponds to non-vanishing D X W, F-term of X uplifts the vacuum energy. non-vanishing D X W, F-term of X uplifts the vacuum energy. Any g(x) is OK, e.g. the Intriligator-Seiberg-Shih model. Any g(x) is OK, e.g. the Intriligator-Seiberg-Shih model.

Simple model For g(X), we use e.g. the ISS model. Intriligator, Seiberg, Shih, ‘ 06 Intriligator, Seiberg, Shih, ‘ 06 The ISS model itself is the global SUSY model. After integrating out heavy modes, we have That breaks SUSY spontaneously One-loop effective potential induces the mass term

Simple model Our model When A=0, SUSY is broken. When A=0, SUSY is broken. We can fine-tune c and μ such that V=0. When μ=0, T is stabilized at the SUSY point,

Simple model Around the reference point, we search the true minimum, where we search the true minimum, where We can do a similar analysis for different g(X). We can also analyze the T-X mixing case We can also analyze the T-X mixing case

Stringy origin for ISS ISS = dual of SU(N) theory with N f flavor quarks, ⇒ Stringy origin Stringy origin Non-perturbative effect Non-perturbative effect

F-term uplifting Three sources of SUSY breaking Modulus mediation and anomaly mediation are Modulus mediation and anomaly mediation are comparable. comparable.

Spectra in F-term uplifting 1) When Fx is sequestered from the visible sector like anti-D3 brane in the original KKLT, like anti-D3 brane in the original KKLT, modulus mediation and anomaly mediation are modulus mediation and anomaly mediation are comparable, i.e. mirage mediation. comparable, i.e. mirage mediation. 2) When contact terms between the visible matter fields and X are not suppressed fields and X are not suppressed sfermion masses ← Fx sfermion masses ← Fx gaugino masses ← mirage mediation gaugino masses ← mirage mediation sfermion masses >> gaugino masses by O(10) sfermion masses >> gaugino masses by O(10)

4.TeV-scale mirage mediation: solution for the fine-tuning problem TeV scale mirage mediation leads to the unique pattern of s-particle spectrum, pattern of s-particle spectrum, gaugino masses degenerate around 1 TeV gaugino masses degenerate around 1 TeV sfermion masses degenerate around 1 TeV sfermion masses degenerate around 1 TeV the little hierarchy between the stop mass the little hierarchy between the stop mass and the Higgs soft mass is naturally realized and the Higgs soft mass is naturally realized at TeV scale. at TeV scale. In this type of models, there is no fine-tuning problem.

4.1 the fine-tuning problem Higgs sector Supersymmetric mass terms in superpotential SUSY scalar potential = (F-term potential) + (D-term potential) Soft SUSY-breaking mass terms No quartic term in soft SUSY breaking terms

Higgs sector (  fine-tuning) The Higgs potential Higgs VEVs

The lightest Higgs mass RG effect

The fine-tuning problem Fine-tuning

Favored pattern of mass parameters If we can realize “ naturally ” the following little hierarchy that would be favorable from a simple that would be favorable from a simple bottom-up viewpoint to avoid the fine-tuning bottom-up viewpoint to avoid the fine-tuning problem. problem. That is possible in the mirage mediation.

4.2 TeV-scale mirage mediation By choosing discrete values, n,m,p,k, we can obtain α=2 ⇒ Mirage scale = TeV scale If the modulus mediation leads to If the modulus mediation leads to the little hierarchy between the stop mass the little hierarchy between the stop mass and the Higgs soft mass is naturally realized and the Higgs soft mass is naturally realized at TeV scale. at TeV scale. at higher loop level at higher loop level In this type of models, there is no fine-tuning problem.

TeV-scale mirage mediation Low-energy spectra (I) (I) (II) They can relax the fine-tuning problem. They have unique patterns. It is interesting to study more their phenomenological aspects.

LSP TeV-scale mirage TeV-scale mirage Higgsisno << bino, wino LSP = Higgsino-like LSP = Higgsino-like Dark matter ??? Thermal relic density is rather small Thermal relic density is rather small

4.3 More about fine-tuning: bottom-up approach Abe, T.K.,Omura, ‘ 07 Abe, T.K.,Omura, ‘ 07 Soft Higgs mass Cancellation between gluino mass and wino mass is important, but bino mass is not so wino mass is important, but bino mass is not so important. important. gluino and wino masses are comparable at Mz. gluino and wino masses are comparable at Mz.

fine-tuning: bottom-up approach Abe, T.K.,Omura, ‘ 07 Abe, T.K.,Omura, ‘ 07 r2 = M2/M3, r1= M1/M3 at Mx r2 = M2/M3, r1= M1/M3 at Mx Favorable region r2=4, -3 < r1 < 10

partial TeV scale mirage unification Abe, Kim, T.K.,Shimizu, ‘ 07 Abe, Kim, T.K.,Shimizu, ‘ 07 Everything is the same as the previous one, except U(1) gauge kinetic function. except U(1) gauge kinetic function. We take U(1) gauge kinetic function different from SU(3) and SU(2) ones such that they lead from SU(3) and SU(2) ones such that they lead to the gauge coupling unification. to the gauge coupling unification.

Extended model LSP = mixture between higgsino and bino ⇒ different phenomenology, ⇒ different phenomenology, e.g. dark matter e.g. dark matter can lead to a right amount of thermal relic density as a dark matter can lead to a right amount of thermal relic density as a dark matter Further study would be interesting Further study would be interesting

Summary We have studied KKLT type models with moduli-mixing superpotential. moduli-mixing superpotential. Soft SUSY breaking terms in generalized models have a rich structure. have a rich structure. in general mixture of modulus mediation and anomaly mediation in general mixture of modulus mediation and anomaly mediation with a certain ratio, i.e. mirage mediation with a certain ratio, i.e. mirage mediation sometimes modulus med. >> anomaly med. sometimes modulus med. >> anomaly med. or modulus med. << anomaly med. or modulus med. << anomaly med.  further study  further study TeV scale partial mirage model also important, e.g. for the fine-tuning problem also important, e.g. for the fine-tuning problem and dark matter physics and dark matter physics

Summary We have also studied on the F-term uplifting scenario, that is, spontaneous SUSY breaking. that is, spontaneous SUSY breaking. Soft SUSY breaking terms in generalized models have a rich structure. have a rich structure. Sfermion masses would be heavier than gaugino masses. Sfermion masses would be heavier than gaugino masses. Sequestering is quite important. Sequestering is quite important.  further study  further study Abe, Higaki, T.K. Omura, in progress Abe, Higaki, T.K. Omura, in progress