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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!
The ground state is where the world resides.
Excited system always come back to the stable ground state and stay there unless excited again.
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…..
Rødovre, near Copenhagen, Denmark
then they are giving up the restrictions of stability ……..
Capital Gate tower of Abu Dhabi ,18 degree lean (4 times that of the Leaning Tower of Pisa)
and start searching for the excitement of the freedom to stay ………
semi-stable
Luckenwalde Town Library (D) near Berlin
The contrast between the stabilty of a classical building and the freedom of a modern structure can be captured here.
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.
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.
1. Fast SUSY primer2. SUSY breaking3. O’Raifeartaigh Model4. Why breaking SUSY is highly restricted5. The ISS proposal Matastable SUSY breaking6. Dienes’ and Thomas’ idea to realize ISS at tree level7. Our simplification of D & T 8. Summary
Contents
Breaking of Supersymmetry by being semi-stable.
Breaking of Supersymmetry by Metastable States.
Symmetry
The difference between them is artificial.
Boson Fermion
222
42 1
Mkkdm
220
2 cmmH
We need a fine-tuning to get a small Higgs mass.
Quadratic Divergence
What’s the motivation?
Hm
SUSY forced the quadratic divergencies in the two loop diagrams to cancel.
Scalar self interaction is related to Yukawa coupling.
Coupling Unification
Extra Space Dimension
String Theory
SUSY algebra
The only non-trivial extension of symmetry in quantum field theory beyond Poincare symmetry and internal symmetry.
This supersymmetry identifies bosons and fermions!
SUSY Semi-Primer
Every Boson comes with a Fermion partner, which behaves identically, and vice versa.
The generators are fermionic (anticommuting, spinorial).
Chiral Superfield combines a scalar φ and a left-handed Weyl spinor ψ
It’s customary to organize SUSY multiplets by superfields: functions of xμ and an imaginary superspace fermionic coordinates θ.
F(x) is a auxiliary field and can be solved in terms other fields by EOM.
Chiral Superfield
Superfield can be expanded in powers of θ. The expansion terminates soon. The components are various ordinary fields in a super-multiplet.
0,,, 2,1,
Just like in tensor analysis, you put SUSY invariant constraints on superfield to get irreducible representations.
022
21
32
!3
1
2
1)( ymW
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.
)(2 Wd
Wess-Zumino Model
Masses of fermions Yukawa coupling between bosons and fermions
General renormalizable SUSY model of Chiral superfields.
W(Φ) controls interactions and masses. Superpotential
Funnily, the integration over θ acts just like differentiation with θ.
i
iF2
Kinetic Energy term comes from ：
iii
ii
WF
)(
It is this scalar potential that will determine the vacuum or vacua.
Equation of Motion solves F completely:
i
iFV2
22 dd
As a bonus, it also gives rise to scalar interactions.
2
2
1 ymFi
4222
4
1 ymV
For Wess-Zumino
Scalar mass and Fermion mass are degenerate.
Scalar self interaction is related to Yukawa coupling.
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.
VV
SUSY vacuum
0iF
SUSY ground state has zero energy!
Take trace
Spontaneous SUSY breaking
0some iF
SUSY will be broken if all the auxiliary fields can not be made zero simultaneously!
Ground state energy is the order parameter.
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.
0F
There is a mass relation for the fields spontaneously breaking SUSY.
Fermions
2
Bosons
2 mm
Spontaneous SUSY breaking can’t be generated in SM or one of the squarks will be too light.
SM
Hidden SUSY
BreakingSectorWeak 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
O’Raifeartaigh (1975) Model (OR model)
Three chiral superfields: 12 ,, X
)()( 12211 gXgW
)( 11 gX
WFX
)( 122
2
gW
F
Generically we can’t make both vanish.
One of the F’s is non-zero. SUSY is borken.
F-type SUSY breaking
X,Φ2 don’t talk to each other.
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.
0some iF
(Consider this as the effective field theory of a more natural UV complete theory)
O’Raifeartaigh Model (OR) (1975)
fhgX
WFX
2111 2
1)(
1122
)(2
mgW
F
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.
For a specific example,
fhg 2111 2
1)( 112 )( mg
)()( 12211 gXgW
The vacuum (vacua) of OR model
minimum conditions
2
21
2
1
22
1
222
2
1
12
mhXmfh
FFFV X
0,1
2
F
V
X
V
02
11
21
21
011
mhfhV
F
01
arbitrary ,02 X0021
11
mhXF
01 arbitrary ,02 X
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 0X
2
21
2
1
22
12
1 mhXmfhV
SUSY is broken by a non-zero F.
0X
We can calculate the masses of scalars and fermions.
hfmhfmmmms 2222 ,,,,0,0
mmm f ,,0
Modulus Fields
SUSY breaking massless Goldstino
At this vacuum
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.
Just like QCD, the fermion components of S forms condensate:
Some hidden chiral superfield S carry the SU(Nc) charge
3SS
This is the F component of the super-meson field SSFSS
It breaks SUSY at scale Λ.
This scale can be naturally small compared to cutoff scale:
cutoffcutoff
ln)(
)(
Mc
M
cutoffcutoff
2
MMe b
On the other hand, FI and OR seems to emerge as the low energy effective theory of a dynamical SUSY breaking mechanism.
1)(
Just like QCD
Coupling Constant α doesn’t need to be fine-tuned to generate a large hierarchy.
Witten Index (1982)
)0()0()()()1(Tr FBE
FBF nnEnEn
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!
0)1(Tr FSUSY 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!
Unfortunately, it doesn’t work!
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.
For these two theories, SUSY is unbroken.
It is possible to calculate Witten index at weak coupling while applying the conclusion to strong coupling.
Witten index is invariant under changes of the Hamiltonian that do not change the far away behavior of the potential!
0)1(Tr FSUSY is unbroken.
SUSY QCD just isn’t like QCD. There is a stable supersymmetric vacuum.
0SS
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.
Affleck, Dine and Seiberg Model has an unbroken U(1)R symmetry. This is a serious problem.
The R charges of the three chiral superfields:
0)(,2)()( 12 RRXR
2. charge is )()( 12211 gXgW
1- charge :)1( RU
Boson and its superpartner have opposite charges.
Superpotential W needs to be charge 2 to preserve U(1)R
U(1)R symmetry
O’Raifeartaigh Model (OR) also has a un unbroken U(1)R symmetry.
)(2 Wd
Vacuum Breaking SUSY)1( RU
Generically it can be proven (Nelson and Seiberg 1994):
Vacuum SUSY)1( RU
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.”
It is really a Fearful Symmetry! Tony Zee
Breaking it by ground state will require an unbearable price.
Enters Meta-Stable Vacua
ISS Model
A supersymmetric SU(Nc) gauge theory with Nf fundamental and anti-fundamental chiral superfields ii QQ ,
iiQmQW
It could be calculated that there is a metastable SUSY breaking vacuum far away from the true vacuum:
The vacuum is supersymmetric.
2mQQIts property can be illustrated by a similer low energy effective theory: OR model.
mcfc NNN2
31
It has a magnetic dual, IR free with a low energy effective theory:
mhW tr~tr
Modify OR model by adding a small mass term for φ2 (Deformation)
1
fhFX 212
1 212 mmF 211
mhXF
Now 3 equations for 3 unknowns, a solution can be found:
This is a SUSY vacuum.
0)(,2)()( 12 RRXRU(1)R has been broken by the small mass term as expected.
01 0,02 X
For small mass, the potential near the previous SUSY breaking minimum is not greatly modified.
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)
01 0,02 X
At A,
At B,
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.
Metastable state breaking SUSY
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.
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.
VV
Try the idea of metastable SUSY breaking by a new mechanism.
The most general supersymmetric Lagrangian of a vector superfield
ja
ijia TgD
In the presence of a chiral field, we can solve the auxiliary D
a
aDV 2
2
1
It gives a scalar potential
a
aD2
2
1
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
kgD iii
In the presence of matter, we can solve the auxiliary D
Fayet-Iliopoulos Term
Under SUSY, F and D change by a total divergence.
Put everything together:
with the all (and the only) important scalar potential:
SUSY vacuum
0,0 DFi
SUSY ground state has zero energy!
Spontaneous SUSY breaking
0or ,0 DFi
SUSY will be broken if all the auxiliary fields can not be made zero simultaneously!
Ground state energy is the order parameter.
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.
0F
Similar for vector superfield:
0D
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 ：
QmQW
kQQQQD
222
8
1kQQQQQmmQV
and a non-zero FI term k for the Abelien gauge theory.
The scalar potential:
If m is large, the minimum is at 0QQ
At this vacuum: 08
1,0 2 kVkD
SUSY is broken by a non-zero D term.
U(1) gauge symmetry is unbroken.
If mass is small km 2
the minimum of V is at kmvvQQ 22with ,0
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.
nest
Achieve a SUSY breaking metastable state perturbatively (tree level calculation).
Dienes and Thomas Model
Try the idea of metastable SUSY breaking by D type model.
The recipe is to put a Wess-Zumino and a Fayet-Illiopoulos together!
Three Chiral Superfields 321 ,,
A Wess-Zumino Superpotential 321 W
Two Abelien U(1) with FI terms: bbbaaa gUgU ,,)1(,,)1(
Massive vector matter with opposite charges in FI
54 , 54mW
Together, you also need to assign appropriate charges to 321 ,,
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!
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.
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.
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:
In the example:
1300B
This is large enough for the metastable lifetime to exceed the age of the universe.
Under certain conditions:
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.
Our Model I: To simplify Dienes & Thomas ModelWe throw away U(1)b
As an example:
We again find structures of minima:
How complicated are we required to be?
A
B
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:
AB
1
3
A
B
C
B
A
C
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. 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.
2. This proposal can be realized at the tree level as suggested by Dienes and Thomas, in a beautiful combination of Wess-Zumino model and Fayet-Illiopoulos Model. Both F-term and D-term acquire non-zero VEV at the metastable local minimum.
3. We simplify this model by reducing the number of U(1) gauge symmetry and superfield and find it works as in DT.
4. Further clarification of why they work and the essence of DT’s proposal is still under investigation.
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