SUSY Breaking by Meta-stable States
Chia-Hung Vincent Chang
NTNU
Based on a work with Kuo-Hsing Tsao now at UIC
NTU theory seminar Dec 2010
Modernism in Architecture
Architecture is about building a residence.
A residence is supposed to project a feeling of home,
stable, stationary, you stay forever once you reach there.
No matter how high reaching the building is,
it always has a stable ground structure
to anchor its excitation.
even a skyscraper……
There are exceptions,
but mostly accidental…..
This stable home feeling ground structure is very
similar to the ground state in modern physics!
Excited system always come back to the stable
ground state and stay there unless excited again.
The ground state is where the world resides.
We think the structure of the stable ground state
will determine how our daily world looks like!
But….
In this ever-changing world, do we really need to
stay at a stable home all the time?
First the architectures of modernism
allow it to flow…..
then they are giving up the restrictions of
stability ……..
Rødovre, near Copenhagen, Denmark
and start searching for the excitement of the freedom to
stay ………
semi-stable
Capital Gate tower of Abu Dhabi ,
18 degree lean (4 times that of the Leaning Tower of Pisa)
The contrast between the stabilty of a
classical building and the freedom of a
modern structure can be captured here.
Luckenwalde Town Library (D) near Berlin
The thing is
If the architectures,
who are supposed to know how to reside
better than we do,
have actually ……
moved on,
to give up total stabilty.
Maybe it is also time for us physcists to
appreciate the thrill and the ecstasy of
being semi-stable.
Breaking of Supersymmetry by being semi-stable.
SUSY Breaking by Meta-stable States
Chia-Hung Vincent Chang
NTNU
Based on a work with Kuo-Hsing Tsao now at UIC
NTU theory seminar Dec 2010
Breaking of Supersymmetry by being semi-stable.
Breaking of Supersymmetry by Metastable States.
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Fast SUSY primer
SUSY breaking
O’Raifeartaigh Model
Why breaking SUSY is highly restricted
The ISS proposal Matastable SUSY breaking
Dienes’ and Thomas’ idea to realize ISS at tree level
Our simplification of D & T
Summary
The difference
between them is
artificial.
Symmetry
Boson
Fermion
What’s the motivation?
Quadratic Divergence

m 
2
1
4 
d
k
  k 2  M
m H  m 0  c
2
2
2

2


2
We need a fine-tuning to get a
small Higgs mass. m H  
SUSY forced the quadratic divergencies in
the two loop diagrams to cancel.
Scalar self interaction is related to Yukawa
coupling.
Coupling Unification
String Theory
Extra Space Dimension
SUSY Semi-Primer
The only non-trivial extension of symmetry in quantum field theory
beyond Poincare symmetry and internal symmetry.
This supersymmetry identifies bosons and fermions!
The generators are fermionic (anticommuting, spinorial).
SUSY algebra
Every Boson comes with a Fermion partner, which behaves identically,
and vice versa.
Chiral Superfield
It’s customary to organize SUSY multiplets by superfields:
functions of xμ and an imaginary superspace fermionic coordinates θ.
  ,       ,       ,     0



 ,   1, 2
1   2  0
2
2
Superfield can be expanded in powers of θ. The expansion terminates
soon. The components are various ordinary fields in a super-multiplet.
Just like in tensor analysis, you put SUSY invariant constraints on
superfield to get irreducible representations.
Chiral Superfield combines a scalar φ and a left-handed Weyl spinor ψ
F(x) is a auxiliary field and can be solved in terms other fields by EOM.
SUSY transformations can be realized as translations in the superspace θ
(plus a translation in space time x).
Given a function of superfield Φ :W(Φ)
SUSY invariants can be obtained by integrating W(Φ) over θ.
Funnily, the integration over θ acts just like differentiation with θ.
 d  W ( ) 
2
Wess-Zumino Model
General renormalizable SUSY model of Chiral superfields.
W ( ) 
1
2
Masses of fermions
m 
2
1
y
3
3!
Yukawa coupling between bosons
and fermions
W(Φ) controls interactions and masses. Superpotential
Kinetic Energy term comes from：  d 2
d    
2



Fi
2
i
As a bonus, it also gives rise to scalar interactions.
Equation of Motion solves F completely:
Fi 
V 

 W ( i )
i
Fi
2

i  i
For Wess-Zumino
Fi  m  
1
2
i
Scalar mass and Fermion mass are degenerate.
Scalar self interaction is related to Yukawa coupling.
It is this scalar potential that will determine the vacuum or vacua.
2
2
V  m  
2
y
1
4
y 
2
4
V

V
Vector Superfield
Vector Superfield combines a vector v and a left-handed Weyl spinor λ.
In Wess-Zumino gauge:
D(x) is a auxiliary field and can be solved in terms other fields by EOM.
SUSY vacuum
Take trace
SUSY ground state has zero energy!
Fi  0
Spontaneous SUSY breaking
Ground state energy is the order parameter.
some
Fi  0
SUSY will be broken if all the auxiliary fields can
not be made zero simultaneously!
Another way to see it:
Spontaneous SUSY Breaking implies that under SUSY transformation:
The transformation of components of a chiral superfield is
The only possible Lorentz invariant non-zero VEV at r.h.s. is that of F.
F 0
There is a mass relation for the fields spontaneously breaking SUSY.

Bosons
m 
2

m
2
Fermions
Spontaneous SUSY breaking can’t be generated in SM or one of the
squarks will be too light.
SM
Hidden
SUSY
Breaking
Sector
Weak Mediating
Sector:
Messenger
The mediation control the phenomenology.
Messenger could be gravity, gauge interaction (SM), anomaly etc.
Messenger can include some SM gauged superfields that also doubly play a role
in breaking SUSY: Direct Gauge Mediation
some
F-type SUSY breaking
Fi  0
O’Raifeartaigh (1975) Model (OR model)
There are as many F-terms as superfield.
In general, there will be a solution for all the F-terms to vanish
unless the superpotential is special-designed.
Three chiral superfields: X ,  2 ,  1
W  Xg 1 ( 1 )   2 g 2 ( 1 )
 FX 
W
X
 g 1 ( 1 )
 F 2 
X,Φ2 don’t talk to each other.
W
 2
 g 2 ( 1 )
Generically we can’t make both vanish.
One of the F’s is non-zero.
SUSY is borken.
(Consider this as the effective field theory of a
more natural UV complete theory)
O’Raifeartaigh Model (OR) (1975)
For a specific example,
g 1 (1 ) 
1
2
g 2 ( 1 )  m  1
h 1  f
2
W  Xg 1 ( 1 )   2 g 2 ( 1 )
 FX 
 F 2 
W
X
W
 2
 g 1 ( 1 ) 
1
2
h 1  f
2
 g 2 (1 )  m  1
These two auxiliary fields are two distinct functions of just one field.
They can’t be zero at the same time.
We can be sure before analyzing the vacuum structure,
SUSY is broken.
The vacuum (vacua) of OR model
minimum conditions
V  FX
1

2
X
,
h  f
V
 2
V
 1
2
 F1
2
2
2
1
2
V
 F 2
F  0
 m 1
2
 hX  1  m  2
2
 F1  0
1

2
2
  h 1  f   h 1  m 1  0
2

1  0
1
 F1  hX  1  m  2
1  0
0
 2  0 , X arbitrary
V 
1
2
2
h 1  f
1  0
2
 m 1
2
 hX  1  m  2
2
 2  0 , X arbitrary
SUSY is broken by a non-zero F.
The non-SUSY vacua is not one state,
but a one complex dimensional space of degenerate.
Pseudo-Moduli Space of Vacua
The degeneracy will be lifted by one-loop effective potential:
The minimum vacuum is at X  0
X 0
At this vacuum
We can calculate the masses of scalars and fermions.
m s  0 , 0 , m , m , m  hf , m  hf
2
2
2
2
Modulus Fields
m f  0, m , m
SUSY breaking massless Goldstino
Dynamical SUSY Breaking
OR models contain scales f that is put in by hand. These scales
generate SUSY breaking scale and may need fine-tuning.
It is natural that we (with Witten) prefer a non-perturbative dynamic
SUSY breaking mechanism where scale are generated by dynamically,
just like Λ in QCD.
A QCD-like SU(Nc) gauge interaction that becomes strong at a scale
Λ.
Some hidden chiral superfield S carry the SU(Nc) charge
Just like QCD, the fermion components of S forms condensate:
 S
S
 
3
This is the F component of the super-meson field
It breaks SUSY at scale Λ.
S S   S
F
S
This scale can be naturally small compared to cutoff scale:
Just like QCD
 ( )
 ( M cutoff )
 e

  c  ln

 ( )  1
M cutoff
2
b
M cutoff  M cutoff
Coupling Constant α doesn’t need to be fine-tuned to generate a
large hierarchy.
On the other hand, FI and OR seems to emerge as the low energy
effective theory of a dynamical SUSY breaking mechanism.
Unfortunately, it doesn’t work!
Witten Index (1982)
Tr (  1)
F


n B ( E )  n F ( E )  n B (0)  n F (0)
E
Every bosonic state of non-vanishing energy pair with
a fermionic state.
If the Witten index is non-zero, there must be a state with
zero energy and hence SUSY is unbroken!
Tr (  1)
F
0
SUSY is unbroken.
(The reverse is not true.)
Witten index is invariant under changes of the Hamiltonian that
do not change the far away behavior of the potential!
Witten index is invariant under changes of the Hamiltonian that
do not change the far away behavior of the potential!
It is possible to calculate Witten index at weak coupling while
applying the conclusion to strong coupling.
Witten index is non-zero for pure SUSY Yang-Mills
theory.
Gauge theories with massive vector-like matter, which flows to
pure Yang-Mills at low energy, will also have a non-zero
Witten indices.
Tr (  1)
F
0
SUSY is unbroken.
For these two theories, SUSY is unbroken.
SUSY QCD just isn’t like QCD. There is a stable supersymmetric
vacuum.
 S
S
0
Four requirements for dynamical SUSY breaking
Chiral Matter
A non-perturbative superpotential generated over the
classical moduli space
A tree level superpotential which completely lifts the
classical moduli space
U(1)R symmetry
(3,2) model by Affleck, Dine and Seiberg (1984)
It contains an U(1)R symmetry.
U(1)R symmetry
Affleck, Dine and Seiberg Model has an unbroken U(1)R symmetry.
This is a serious problem.
U (1) R :  charge - 1
Boson and its superpartner have opposite charges.
Superpotential W needs to be charge 2 to preserve U(1)R  d  2 W ( )
O’Raifeartaigh Model (OR) also has a un unbroken U(1)R symmetry.
The R charges of the three chiral superfields:
R ( X )  R (  2 )  2 , R ( 1 )  0
W  Xg 1 ( 1 )   2 g 2 ( 1 ) is charge 2.
Generically it can be proven (Nelson and Seiberg 1994):
U (1) R  SUSY Breaking
U (1) R  SUSY
Vacuum
Vacuum
An unbroken U(1)R symmetry will prohibit Majorana
gaugino masses (R invariant) and render model-building very
difficult.
SUSY breaking seems to be a rather non-generic phenomenon.
“The issue of SUSY breaking by ground state has a topological nature:
it depends only on asymptotics and global properties of the theory.”
Breaking it by ground state will require an unbearable price.
It is really a Fearful Symmetry! Tony Zee
Enters Meta-Stable Vacua
ISS Model
A supersymmetric SU(Nc) gauge theory with Nf fundamental
and anti-fundamental chiral superfields Q i , Q i
W  mQ i Q i
The vacuum is supersymmetric.
Nc 1 N
f

3
2
Nc
m  
It has a magnetic dual, IR free with a low energy effective theory:
W  h tr   ~    tr m 
It could be calculated that there is a metastable SUSY
breaking vacuum far away from the true vacuum:
 Q Q  m 
2
Its property can be illustrated by a similer low energy effective
theory: OR model.
Modify OR model by adding a small mass term for φ2 (Deformation)
  1
 FX 
1
2
h 1  f
2
 F 2  m  1   m  2
 F1  hX  1  m  2
Now 3 equations for 3 unknowns, a solution can be found:
This is a SUSY vacuum.
U(1)R has been broken by the small mass term as expected.
R ( X )  R (  2 )  2 , R ( 1 )  0
For small mass, the potential near the previous SUSY breaking minimum
is not greatly modified.
1  0  2  0 , X  0
It will still be locally stable. Hence it becomes a metastable vacuum!
The universe can live in the metastable vacua with SUSY broken.
Globally, there is a SUSY vacuum, hence ensuring U(1)R is broken.
Using metastable state to break SUSY while keeping a SUSY ground
state could help evade a lot of constraints such as Witten Index.
“Breaking SUSY by long-living metastable states is generic.”
Intrilligator, Seiberg, Shih (2006)
At A,
At B,  1  0  2  0 , X  0
Metastable state breaking SUSY
As ε becomes smaller, SUSY vacuum A will be pushed further and
further, diminishing the tunneling rate as small as you like, until
disappear into infinity at ε=0.
With a SUSY vacuum, R symmetry is explicitly broken and gaugino
masses are generated.
However the small parameter also render the gaugino masses very small.
Try the idea of metastable SUSY breaking by a new mechanism.
V

V
Vector Superfield
Vector Superfield combines a vector v and a left-handed Weyl spinor λ.
In Wess-Zumino gauge:
D(x) is a auxiliary field and can be solved in terms other fields by EOM.
The most general supersymmetric Lagrangian of a vector superfield


1
D

2
a
In the presence of a chiral field, we can solve the auxiliary D
D
a

 g  i T ij 
a
It gives a scalar potential
V 
1
2

a
2
Da
j
2
a
Under SUSY, F and D change by a total divergence.
For Abelien gauge theory, the D term of a vector superfield
is both gauge invariant and supersymmetric.
We can add a D-term to the Lagrangian:
kD
Fayet-Iliopoulos Term
In the presence of matter, we can solve the auxiliary D

D  g i i  i  k
Put everything together:
with the all (and the only) important scalar potential:
SUSY vacuum
SUSY ground state has zero energy!
Fi  0 ,
D0
Spontaneous SUSY breaking
Ground state energy is the order parameter.
Fi  0 ,
or
D0
SUSY will be broken if all the auxiliary fields can
not be made zero simultaneously!
Another way to see it:
Spontaneous SUSY Breaking implies that under SUSY transformation:
The transformation of components of a chiral superfield is
The only possible Lorentz invariant non-zero VEV at r.h.s. is that of F.
F 0
Similar for vector superfield:
D 0
D-type SUSY breaking
Fayet-Illiopoulos mechanism (1974)
Assuming an Abelien Gauge Theory:
Two Chiral Superfield Q, Q with opposite charge +1, -1
Introduce a non-zero mass for Q：
W  mQ Q
and a non-zero FI term k for the Abelien gauge theory.


D Q Q Q Q k
The scalar potential:
V  mQ
2
 mQ
2

1

2

Q Q Q Q k
8
If m is large, the minimum is at Q  Q  0 U(1) gauge symmetry is unbroken.
D  k  0, V 
1
k 0
2
At this vacuum:
8
SUSY is broken by a non-zero D term.
If mass is small m 2  k
the minimum of V is at
Q  0, Q  v
with
v m k
2
2
SUSY is broken by non-zero D term and F terms.
U(1) gauge symmetry is now broken
We expect a massive gauge boson and massless goldstino
(mixture of gaugino and fermionic component of Q) of SUSY
breaking.
Try the idea of metastable SUSY breaking by D type model.
Dienes and Thomas Model
nest
Achieve a SUSY breaking metastable state
perturbatively (tree level calculation).
The recipe is to put a Wess-Zumino and a Fayet-Illiopoulos together!
Three Chiral Superfields  1 ,  2 ,  3 
A Wess-Zumino Superpotential
W    1 2  3
Two Abelien U(1) with FI terms: U (1) a ,  a , g a
U (1) b ,  b , g b
Massive vector matter with opposite charges in FI
 4 ,  5 
W  m 4 5
Together, you also need to assign appropriate charges to  1 ,  2 ,  3 
The extrema are determined by the conditions:
Solutions is a local minimum if the following mass matrix
contains only positive eigenvalues! This is the hard part!
As an example, choose
A is a SUSY true vacuum, with R
symmetry and a U(1) gauge symmetry.
B is a SUSY breaking metastable local minimum,
with broken R symmetry and broken U(1) gauge
symmetry.
A is a SUSY true vacuum, with R symmetry and a U(1) gauge symmetry.
B is a SUSY breaking metastable local minimum, with broken R
symmetry and broken U(1) gauge symmetry.
This model has the benefits:
It’s tree level mechanism and everything is calculable.
No small parameters are needed. The metastable states are
produced and sustained by the complexity of the model.
Lifetime of the metastable state
The metastable state tunnels to the true vacuum through instanton
transition.
The decay rate per unit volume is
B is calculated from the distances in field space between
barrier top (C) and metastable state (B) or true vacuum (A):
and the potential differences between similar combinations:
Under certain conditions:
In the example:
B  1300
This is large enough for the metastable lifetime to exceed the age of
the universe.
A is a SUSY true vacuum, with R symmetry and a U(1) gauge symmetry.
B is a SUSY breaking metastable local minimum, with broken R
symmetry and broken U(1) gauge symmetry.
This model has the benefits:
No small parameters are needed. The metastable states are
produced and sustained by the complexity of the model.
How complicated are we required to be?
Our Model I: To simplify Dienes & Thomas Model
We throw away U(1)b
As an example:
We again find structures of minima:
B
A
The metastable minimum is a bit shallow!
It will decrease the lifetime of B, but it turns out still OK.
B
Our Model II: we simplify our Model I even further:
We throw away one superfield and U(1)b
As an example:
We again find structures of minima:
A
B
3
A
C
B
1
C
B
A
We have constructed a model which is one field and one
Abelien Gauge symmetry short of the Dienes Thomas
Model, but achieves the same ground state structure.
The metastable local minimum is about as deep in DT
and the distance between A,B is also about the same
order. We expect the lifetime of metastable to exceed
the age of the universe.
How about a model without mass terms?
Summary
1.
2.
3.
4.
Breaking SUSY by metastable states and allowing at the
same time a SUSY vacuum let model building escape from
the stringent constraints posed by the global nature of SUSY
breaking. It becomes generic and easy to build.
This proposal can be realized at the tree level as suggested
by Dienes and Thomas, in a beautiful combination of WessZumino model and Fayet-Illiopoulos Model. Both F-term
and D-term acquire non-zero VEV at the metastable local
minimum.
We simplify this model by reducing the number of U(1)
gauge symmetry and superfield and find it works as in DT.
Further clarification of why they work and the essence of
DT’s proposal is still under investigation.