Metastable supersymmetry breaking
vacua from conformal dynamics
Yuji Omura
(Kyoto University)
Based on
Hiroyuki Abe, Tatsuo Kobayashi, and Yuji Omura,
arXiv:0712.2519 [hep-ph] .
1. Introduction
We suggest the scenario that conformal dynamics
leads to metastable supersymmetry breaking vacua.
First, let me introduce the general argument about SUSY breaking shortly.
The argument about SUSY breaking
Based on the Nelson-Seiberg argument, the models which cause SUSY
breaking have U(1)R symmetry,
   ei ,W  W e2i .
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.
The generalized O’Raifeartaigh model is written down as
NX
WOR ( X a , i )   X a g a (i )
a 1
where
(NX>Nφ )
X a : R[ X a ]  2, i : R[i ]  0,
g a   are generic functions .
The F-flat conditions of Xa are
 X a W  g a (i )  0.
(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).
WOR  WR  X a g a    mab X a X b  abc X a X b X c  w 
These R-symmetry breaking terms make SUSY vacua appear
and SUSY breaking vacua disappear.
How can we realize SUSY breaking vacua?
V
One-loop effective potential can stabilize a SUSY
breaking vacuum near the origin.
2
2
V  VSUSY  m X
X
X
On the other hand, the R-symmetry breaking
terms destabilize the SUSY breaking vacuum.
WOR  WR  X a g a    mab X a X b  abc X a X b X c  w 
If they are much smaller than the loop effect,
a metastable SUSY breaking vacuum can be realized.
w( ) would also lead to SUSY vacua, but they disappear under the limit, mab , abc  0 .
, 0
 X aW  ga ( )  mab X b  abc X b X c m
 ga ( )

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
3N  N f 
3
N
2
which corresponds to the conformal window.
•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 model
Chiral matter fields are
~
ia , ia ,  ij
(i  1, , N f , a, a  1, , N ).
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.
SU(N)
 ij
ia
~
ia
a
W

X
g a (I )
SU(Nf)
OR
1
adj
Xa
N
Nf
I
N
Nf
I
The superpotential without R-symmetry, at the renormalizable level, is
~
~
2
3
W  hTrN f [ ]  f TrN f []  mTrN f [ ]  TrN f [ ]  m TrN f [ ].

Vacuum structure
~
~
2
3
W  hTrN f [ ]  f TrN f []  mTrN f [←
 corresponds
]  TrN f [to] 
Wm
OR Tr
. N f [ ]
~
~
 Y Z       ~    
(


)ij : rank ( N )

   ~
~


 
 
( ) ij : rank ( N f )
Z X 
If R-symmetry is preserved, there is a SUSY breaking vacuum,
 a ~b  
X : flat
f
 ab , Yab  0
h
V  N
f
 N f
2
m , 0
 
(This SUSY breaking vacuum corresponds to the solution in the ISS model.)
If R-symmetry is not preserved,
the SUSY breaking vacuum is destabilized and SUSY vacua appear.
SUSY vacua
 a ~b  
X IJ 
f
 ab , Yab  0
h
m
m 2  4 f
 IJ
2
, 0
m
 
Effective Lagrangian with cut-off
3N  N f 
3
N ,which corresponds to the conformal window, is
2
satisfied in our model, so that gauge coupling and yukawa coupling
have fixed-points.
Gauge coupling fixed-point
2
3N  N f  N f   
  
1  N
  
0
3N  N f
Nf
0
1     0
Yukawa coupling fixed-point
 h  h       ~   0
   2   0
0    2
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.
~
~
2
3
W  hTrN f [ ]  f TrN f []  mTrN f [ ]  TrN f [ ]  m TrN f [ ]
“the R-symmetric superpotential”
Near the fixed-point, these R-symmetry breaking terms are estimated as,




1

f      f  
f    Z   2 f  
1

2
Z   


2 



1
m    
m   m( )

m   Z   m 


    Z  
3
2

  
    


 
3 
     ( )
It is important that the suppression of f is the weakest. This means even if
we assume f ()  m2 () , this superpotential approximates “the R-symmetric
superpotential”, which causes SUSY breaking, at low energy scale.
mφ has a negative anomalous dimension, so that mφ becomes enhanced ,

m     

2 
m  
  
We will comment on such terms later.
3N  N f
Nf
 0.
V
Loop effect
X
 0
We set the parameters at

2
2
2
f
(

)

m
(

)


,
m
()  f ().

as follows:
SUSY vacua are estimated as,
X IJ 
m
m 2  4 f
 IJ
2
X SUSY

  

 0


Under the limit,   0 , these SUSY vacua go far away from X=0.
A metastable SUSY breaking vacuum appear, when the potential of X becomes
enough flat for the one-loop mass to be efficient .
V  VSUSY  m X2 X
2
h3 f
m  2 N N f  N log 4  1
8
2
X
The solution is
X 
fm   
 
mX2

2 
.
This solution cannot be
defined under the limit,
  .
How long does the conformal dynamics need to last
to realize the SUSY breaking vacuum?
If we assume f ()  m2 ()  2 , m2 ()  f (),
the supersymmetric mass and the one-loop mass are estimated as
m()  m X  O(10 3 / 2 ) h 3 f () .
The supersymmetric mass is suppressed by conformal dynamics
 
m X    X 
  
2 
 
m   m X  X   O (10 3 / 2 ) hf   X 
  
 /2
.
The scale, where the supersymmetric mass becomes the same order
as the one-loop mass, is
 X 
 

3  / 2
 O(103 / 2 ).
0    2
Nf 
19
1


,


10
GeV ,

For example, in the case
n
19 n
the one-loop mass becomes important below  X  10 GeV .
SUSY can be broken below μX.
3
N
2
In the region ( m <<mX ), the F-component of  ijis estimated
as O( f (  )) , so that the SUSY breaking scale (Λint) is
 f   
 int     
  
1
( 2   )

int
O


f  int  .
However,
~
~
w( ,  )  m TrN f [ ]
If the φ mass term,
this SUSYbreaking scale would change.
, is as large as the other terms,
~
This is because the anomalous dimension of  ,  is negative,
2

3N  N f

m      m  
  
 0.

N
 
f
This term becomes bigger at low energy, so that this theory gets out of the
confomal window at the scale,
 D  m  D ,
~
where  ,  decouple with other fields .
If m  int  
f  int  is satisfied, SUSY breaking scale changes.
Furthermore, in conformal dynamics ,
the strong coupling anomalous dimension can suppress FCNC.
Lhid vis M p  
cij
M pn  2
Qi Q j Ohid  
Ref)M. A. Luty and R. Sundrum,
PRD65 ,066004(2002),PRD67,045007(2003)
anomalous dimension
of Ohid(Φ).

cij  E 
 Qi Q j Ohid  
Lhid vis E   n  2 

M p  M p 
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
mij  int  
2
FX
M
2
2
p
  int

 Mp



 ,


gaugino mass term
F   int
M a  int   X 
M p  M p





2
.
0    2
3.5D model
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,
( y, x)  ecy ( x)


 cR
c: kink mass
  e

,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.
W 
( )
(0)
dy
{

(
y


)
W


(
y
)
W
}

W ( )  e  c X Ry fX  hij e
 ( c X  ci  c j ) Ry
X i j  e
 ( ci  c j ) Ry
mij i j  e 2c X Ry mX 2
( 0)
W ( 0)  h11
X12


W  h11( 0)  h11e  c X R X12  e  c X R  fX  m1212   e 2 c X RWR
4. Summary
We argued SUSY breaking in the generalized O’Raighfeartaigh Model,
WOR  WR  X a g a    mab X a X b  abc X a X b X c  w 
Xa are the flat directions.
These terms destabilize
SUSY breaking vacuum.
One-loop effective potential
stabilizes SUSY breaking vacua.
V  VSUSY  m X2 X
The coefficients of squared X and cubed X need to be
suppressed, compared with the the coefficients of X.
Conformal dynamics
2
In the limit,
mab , abc  0,
these terms don’t
make SUSY vacua
appear.
If the loop effect is bigger
than the R-symmetry
breaking terms, SUSY can
be broken.
We discussed the SU(N) gauge theory with Nf flavor which has an IR fixed-point.
3
3
N

N

N,
f
The number of flavor satisfies
2
which corresponds to the conformal window.
~
~
W  hTrN f [ ]  f TrN f []  mTrN f [ 2 ]  TrN f [ 3 ]  m TrN f [ ]
3
h
f
2
m

N N f  N log 4  1
V  VSUSY  m X X
2
8
High energy scale ( Λ )
If we assume f ()  m2 ()  2 , m2 ()  f (),
SUSY is preserved because
the R-symmetry breaking terms are too large, compared with the mass term in
the one-loop effective potential.
2
2
X
m()  mX ()
Low energy scale
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  .
Z
1
2


 


How long does the conformal dynamics need to last
to realize the SUSY breaking vacuum?
•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    1 ,   1019 GeV , the scale where the one-loop effective
n
potential becomes efficient is estimated as
 X  1019n GeV ,
( f ()  m2 ()  2 , m2 ()  f ()).
•The SUSY breaking scale, the nonzero F-component, is
f int  
int 
~
f   1010  GeV



 O(101 ) .
~
•If mφ , w( ,  )  m TrN f [ ], is large, compared with other terms,
this theory removes away from the conformal windows
~
at the scale,  D  m   ,where  ,  decouple with other fields.
In this case, SUSY breaking scale would change.
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.
END