Metastable supersymmetry breaking Metastable supersymmetry breaking vacua from conformal dynamicsvacua from conformal dynamics

Based on Hiroyuki Abe, Tatsuo Kobayashi, and

Yuji Omura, arXiv:0712.2519 [hep-ph] .

Yuji Omura　 (Kyoto University)

1. Introduction1. IntroductionWe suggest the scenario that conformal dynamics leads to metastable supersymmetry breaking

vacua.

., 2 ii eWWe

The argument about SUSY breaking Based on the Nelson-Seiberg argument, the models which cause SUSY breaking have U(1)R symmetry,

For examlpe, the O’Raifeartaigh model, which we discuss in this talk, is known as one of the models which cause SUSY breaking, and it has R-symmetry.

First, let me introduce the general argument about SUSY breaking shortly.

The generalized O’Raifeartaigh model is written down as

)(),(1

ia

N

aaiaOR gXXW

X

,0][:,2][: iiaa RXRX (NX>Nφ )

.0)( iaX gWa

(NX-Nφ) Xa are flat directions.

They don`t have solutions, because they are NX (>Nφ

)equations with Nφ unknowns.

On the other hand, R-symmetry must be broken explicitly to get nonzero gaugino masses, and to avoid massless boson (Goldstone boson).

wXXXXXmgXWW cbaabc

baab

aaROR

These R-symmetry breaking terms make SUSY vacua appear and SUSY breaking vacua disappear.

where are generic functions .

The F-flat conditions of Xa

are

ag

wXXXXXmgXWW cbaabc

baab

aaROR

22 XmVV XSUSY V

X

One-loop effective potential can stabilize a SUSY breaking vacuum near the origin.

If they are much smaller than the loop effect,a metastable SUSY breaking vacuum can be realized. 　　　 would also lead to SUSY vacua, but they disappear under the

limit, . )(w

)()( 0, a

mcbabc

babaX gXXXmgW

a

0, abcabm

How can we realize SUSY breaking vacua?

On the other hand, the R-symmetry breaking terms destabilize the SUSY breaking vacuum.

We suggest conformal dynamics to realize enough small mab

λabc .

The features of our model

•Our model is the SU(N) gauge theory with Nf flavors.•The flavor number, Nf, satisfies

which corresponds to the conformal window.

NNN f 2

33

•Our model doesn’t have R-symmetry and its superpotential is generic at the renormalizable level. •The dynamics can lead to conformal sequestering.

2. 4D conformal model2. 4D conformal model).,,1,,,,1(,

~, NaaNi fijaiia

].~

[][][][]~

[ 32 fffff NNNNN TrmTrmTrTrfTrhW

Chiral matter fields are

SU(N)

SU(Nf)

ij

ia

ia~

)( Iaa

OR gXW

N

N

fN

fN

1 adj aX

I

I

SU(N) symmetry is imposed on these fields as follows, and each field corresponds to φ , X in the Generalized O’Raifeartaigh Model in the introduction. Furthermore, we impose SU(Nf) flavor symmetry to make the analysis easier, but the following discussions would be valid, even if the flavor symmetry is explicitly broken.

The superpotential without R-symmetry, at the renormalizable level, is

Vacuum structure

SUSY vacua

XZ

ZY~

~

~~

IJIJ

fmmX

2

42

0,~ ababba Yh

f 2fNNV f

0,~ ababba Yh

f

X : flat

0,m

)(:)~

( Nrankij)(:)( fij Nrank

If R-symmetry is preserved, there is a SUSY breaking vacuum,

If R-symmetry is not preserved, the SUSY breaking vacuum is destabilized and SUSY vacua appear.

(This SUSY breaking vacuum corresponds to the solution in the ISS model.)

0,m

]~

[][][][]~

[ 32 fffff NNNNN TrmTrmTrTrfTrhW ← corresponds to WOR .

Effective Lagrangian with cut-off

Gauge coupling fixed-point

ff NNNN

31

20

3

f

f

N

NN

Yukawa coupling fixed-point

~ hh

02

20

01

NNN f 2

33 ,which corresponds to the conformal window, is satisfied in our model, so that gauge coupling and yukawa coupling have fixed-points.

0

0

This theory is completely conformal at the fixed-point , so that SUSY would not be broken there. We suggest there is a parameter region which causes SUSY breaking near the fixed-point.

ff

fZf 21

mZm 1

2

3Z

3

mm2

21

Z

)(m

)(

Near the fixed-point, these R-symmetry breaking terms are estimated as,

]~

[][][][]~

[ 32 fffff NNNNN TrmTrmTrTrfTrhW

mφ has a negative anomalous dimension, so that mφ becomes enhanced ,

We will comment on such terms later.

mm2

.03

f

f

N

NN

It is important that the suppression of f is the weakest. This means even if we assume , this superpotential approximates “the R-symmetric superpotential”, which causes SUSY breaking, at low energy scale.

)()( 2 mf

“the R-symmetric superpotential”

X

22 XmVV XSUSY

14log8 2

32 NNN

fhm fX

A metastable SUSY breaking vacuum appear, when the potential of X becomes enough flat for the one-loop mass to be efficient .

SUSY vacua are estimated as,

.2

2

Xm

fmX

SUSYX

V

).()(,)()( 222 fmmf

IJIJ

fmmX

2

42

Loop effect

0

0

.

We set the parameters at as follows:

Under the limit, , these SUSY vacua go far away from X=0.

The solution is This solution cannot be defined under the limit,

0

For example, in the case , the one-loop mass becomes important below . SUSY can be broken below μX.

How long does the conformal dynamics need to last to realize the SUSY breaking vacuum?

.)10(2/

2/32

X

XXX

X hfOmmm

.)()10()( 32/3 fhOmm X

),()(,)()( 222 fmmf If we assume 　　　　　　　　　　　　　　　　the supersymmetric mass and the one-loop mass are estimated as

The supersymmetric mass is suppressed by conformal dynamics

).10( 2/32/3

OX

20

GeVn1910,1

NN f 2

3

GeVnX

1910

The scale, where the supersymmetric mass becomes the same order as the one-loop mass, is

If is satisfied, SUSY breaking scale changes.

This term becomes bigger at low energy, so that this theory gets out of the confomal window at the scale,

where decouple with other fields .

mm2

]~

[)~,( fNTrmw If the φ mass term, , is as

large as the other terms,this SUSYbreaking scale would change.

,DD m

.03

f

f

N

NN

This is because the anomalous dimension of is negative,

intint fm

In the region ( m <<mX ), the F-component of is estimated as , so that the SUSY breaking scale (Λint) is

))(( fO

However,

)2(1

int

f .intint fO

ij

~,

~,

the strong coupling anomalous dimension can suppress FCNC.

Furthermore, in conformal dynamics ,

hidjinp

ijpvishid OQQ

M

cML

2

hidji

pnp

ijvishid OQQ

M

E

M

cEL

2

anomalous dimensionof Ohid(Φ).

,int2

2

int2

pp

Xij MM

Fm

.2

intint

pp

Xa MM

FM

In our model the F-component of is nonzero, so the direct couplings of with the visible sectors are suppressed by the same order as the R-symmetry breaking sectors : scalar mass

term

gaugino mass term

20

Ref)M. A. Luty and R. Sundrum,

PRD65 ,066004(2002),PRD67,045007(2003)

3.5D model3.5D model

)(),( xexy yc

We can construct simply various models within the framework of 5D orbifold theory . Renormalization group flows in the 4D theory correspond to exponential profiles of zero modes,

RRcRcRc WemfXeXehhW XXX

2

21122111

)0(11

})()({ )0()( WyWydyW 22)()()( mXemeXehfXeW Rycji

ijRyccjiRyccc

ijRyc XjijiXX

21

)0(11

)0( XhW

,where R is the radius of the fifth dimension, y and c is the constant which do not have constraints. For example, we consider the 5D theory whose 5-th dimension is compactified on S1/Z2. If we suppose that the following superpotential is allowed on the fixed-points, SUSY breaking is realized.

Rce

c: kink mass

In the limit,

these terms don’t make SUSY vacua appear.

4. Summary 4. Summary We argued SUSY breaking in the generalized O’Raighfeartaigh Model,

wXXXXXmgXWW cbaabc

baab

aaROR

Conformal dynamics

Xa are the flat directions.

One-loop effective potential stabilizes SUSY breaking vacua.

22 XmVV XSUSY

These terms destabilize SUSY breaking vacuum.

If the loop effect is bigger than the R-symmetry breaking terms, SUSY can be broken.

The coefficients of squared X and cubed X need to be suppressed, compared with the the coefficients of X.

,0, abcabm

If we assume SUSY is preserved because the R-symmetry breaking terms are too large, compared with the mass term in the one-loop effective potential.

The number of flavor satisfies which corresponds to the conformal window.

A metastable SUSY breaking vacuum appears when R-symmetry breaking terms are suppressed, compared with mX. The suppression is caused by the positive anomalous dimension of .

]~

[][][][]~

[ 32 fffff NNNNN TrmTrmTrTrfTrhW

We discussed the SU(N) gauge theory with Nf flavor which has an IR fixed-point. ,

2

33 NNN f

High energy scale ( Λ )

Low energy scale

14log8 2

32 NNN

fhm fX

)()( Xmm

22 XmVV XSUSY

),()(,)()( 222 fmmf

21

Z

•If mφ , is large, compared with other terms, this theory removes away from the conformal windows at the scale, ,where decouple with other fields.

How long does the conformal dynamics need to last to realize the SUSY breaking vacuum?

],~

[)~,( fNTrmw

mD ~,

•It depends on N and Nf. If Nf is close to 3N, is so small that the flow has to be as long as possible. In the case , the scale where the one-loop effective potential becomes efficient is estimated as 　　　

•The SUSY breaking scale, the nonzero F-component, is

This scenario can lead to conformal sequestering.We suggest the construction within the framework of 5D theory according to the correspondence between Renormalization group flows in the 4D theory and exponential profiles of zero modes.

In this case, SUSY breaking scale would change.

END

GeVn1910,1

,1019 GeVnX

GeVff 10intint 10 .)10( 1

O

)).()(,)()(( 222 fmmf