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Meta-stable Vacua in
SQCD and MQCD
David Shih
Harvard University
K. Intriligator, N. Seiberg and DS
hep-th/0602239
I. Bena, E. Gorbatov, S. Hellerman, N. Seiberg and DS
hep-th/0608157
Field theory motivation
Dynamical SUSY breaking (DSB) is the idea that
supersymmetry is spontaneously broken through nonperturbative effects in an asymptotically free gauge theory.
It provides a natural mechanism for generating large mass
hierarchies (Witten). In fact, it is the only mechanism of
natural SUSY breaking we know of.
As such, it is directly relevant to particle phenomenology
and supersymmetric model building.
Field theory motivation (cont’d)
However, realistic SUSY models tend to be complicated
constructions with many components.
Hidden
Sector (DSB)
Messenger
sector
Visible sector
(e.g. MSSM)
Part of the complexity stems from the dearth of simple
examples of DSB in field theory.
String theory motivation
Existing non-SUSY 4D string compactifications also tend to
be quite complicated.
Part of the complexity stems from the dearth of simple
stable SUSY breaking compactifications in string theory.
Can these constructions be simplified??
DSB, simplified
Recently a particularly simple class of field theories was
shown to exhibit (meta-stable) DSB:
N=1 SQCD with massive flavors in the free-magnetic phase.
Evidently, DSB is a more generic phenomenon than
previously thought. Hopefully this will lead to much simpler
low-energy models.
Can embedding this in string theory teach us anything about
SUSY breaking in string theory?
Can it lead to simpler SUSY string compactifications?
Outline
• Part I: Meta-stable vacua in SQCD.
– SUSY vacua of the electric theory.
– Non-SUSY vacua of the magnetic theory.
– Seiberg duality and meta-stable DSB.
• Part II: Meta-stable vacua in MQCD?
–
–
–
–
Embedding into IIA string theory using NS5, D4, D6 branes.
SUSY electric brane configurations.
Non-SUSY magnetic brane configurations.
An obstruction in the lift to M-theory…
Part I:
Meta-stable Vacua in SQCD
The “Electric” Theory
• Beta function:
Asymptotically free if Nf < 3Nc
• Superpotential:
• SUSY vacua:
h Qi = h Q
~i = 0
• Perturbative:
hM i = (¤3Nc ¡Nf mNf ¡Nc )1=Nc
• Exact:
0
The “Magnetic” Theory
• Beta function:
Asymptotically free if Nf > 3Nc/2
(IR free if not)
• Superpotential:
• SUSY vacua:
• Perturbative:
SUSY
vacua!
¡
hM i = no
¡
(¤3Nc Nf mNf Nc )1=Nc
0
• Exact:
Let us understand this in more detail…
“Rank condition” SUSY breaking
SUSY is broken at tree-level:
(analogous to the O’Raifeartaigh model)
(rank Nf -Nc )
(rank Nf )
(Of course, this SUSY breaking is a check of the duality.
Otherwise, we would have had extra SUSY vacua.)
Focus on the free-magnetic phase
What is the vacuum structure?
Let’s focus now on
where theory is IR free.
Then the Kahler potential is smooth near the origin:
Using this, we can compute the scalar potential near the origin:
Non-SUSY vacua of the magnetic theory
Classical vacua (up to global symmetries):
Arbitrary
matrix
(Note:
matrices
)
Potential for the pseudo-moduli
and
parameterize a pseudo-moduli space.
This space is lifted in perturbation theory, since SUSY
is broken.
Potential for the pseudo-moduli, cont’d
In fact, expanding around the point of maximal unbroken global
symmetry:
and using the Coleman-Weinberg formula, we find
The mass-squareds are all positive!
For
, the magnetic theory has local
SUSY-breaking vacua!
Some properties of the SUSY vacua
• Global symmetries:
• Massless modes:
– Goldstino
– Goldstone bosons
– gauginos and other fermions from unbroken discrete
• Vacuum energy:
Seiberg Duality
What does this have to do with the “electric”
theory?
SQCD
Need Seiberg duality – “Electric” and “magnetic” theories
become the same theory in the IR.
.
Electric
.
Magnetic
“Free magnetic phase”
Magnetic theory is an IR free effective description of the
electric theory at energies
!
Meta-stable DSB in SQCD
Thus there is a meta-stable SUSY-breaking
vacuum in SUSY QCD!
•
Can show that it is parametrically long-lived for
•
Can show that higher loops, non-perturbative effects, and Kahler
potential corrections are not important near the origin for
Nc SUSY vacua
Effect of
Part II:
Meta-stable Vacua in MQCD?
What is MQCD?
• It is a continuation of SUSY gauge theories into brane
configurations of IIA string theory and M-theory.
• Typically there is a length scale
which, along
with
and , controls whether we are in the (nonoverlapping) field theory, IIA, or M-theory regime.
Magnetic IIA brane
configuration
MQCD
MQCD
SQCD
Electric IIA brane
configuration
MQCD (cont’d)
• Due to holomorphy, MQCD can reproduce detailed
supersymmetric properties of many N=1 and N=2 gauge
theories. This often leads to nice geometric
interpretations of these theories.
• Although non-supersymmetric properties are known to
quantitatively disagree between MQCD and the
corresponding field theories, it was widely hoped that the
two would at least qualitatively agree.
• We will show that in fact, MQCD and SQCD have
important qualitative differences – the latter has metastable vacua that the former does not.
The “Electric” Brane Configuration
We will be interested in the IIA brane construction of N=1
SQCD using NS5, D4 and D6 branes:
(Elitzur, Giveon & Kutasov)
Field theory limit:
(In this limit, the brane
description is not valid.)
Brane Bending
At gs 6= 0, the NS5 branes bend due to their interactions
with the D4 and D6 branes.
Mass deformation
Giving the electric quarks masses corresponds to moving the D6
branes in the
direction. The resulting configuration is
still supersymmetric.
Also, the bending is unchanged:
The “Magnetic” Brane Configuration
The “magnetic theory” has a similar IIA brane construction…
…with the same bending at infinity.
Magnetic mass deformation
In the magnetic brane configuration, the analogue of the mass
deformation appears to break supersymmetry.
This configuration
corresponds to origin
of field space in the
magnetic theory.
Mass deformation (cont’d)
The system can lower its energy by snapping together
of the D4 branes.
This provides a simple geometric realization
of the SUSY-breaking vacuum of the
magnetic theory. (Notice, however, that the NS’
bending is different now…)
Holomorphic M5 branes
But is this the IIA lift of the meta-stable vacuum of SQCD?
For that, we need to understand the
SUSY
magnetic brane configuration. It exists due to nonperturbative effects in M-theory.
It can be thought of as a smooth M5 brane wrapping the
holomorphic curve (Hori et al., Brandhuber et al.)
where
and
Limits of the holomorphic M5 curve
(Recall that
)
Meta-stable non-holomorphic curve?
If there is an MQCD lift of the SQCD meta-stable state, it
must be a non-holomorphic, minimal-area surface with the
same behavior at infinity as the SUSY curve.
In particular, it must become the NS’ brane as
with the bending we saw above:
No solution
In fact, with a plausible ansatz and a lot of calculation, one
can show that the any solution satisfying the boundary
conditions at infinity must have p
m ¡ cx6 + c (x6 )2 ¡ 2cmx6 + m2
x4 + ix5 =
1 ¡ c2
v(x6 = 0) =
m
1¡c
But then
intersect the D6 brane!
6= 0
and the curve does not
So there is no solution! The meta-stable state of SQCD
does not have a lift to MQCD. (Indeed, this should have been
expected from the bending of the IIA brane configuration.)
Interpretation
What does this mean? Evidently, the meta-stable state of
SQCD develops an instability at some point during the
continuation to MQCD.
This is surprising, because the instability does not correspond
to any of the light field theory modes. Instead, it must be a
new mode on the M5 brane which does not decouple.
We conclude that the non-supersymmetric data of SQCD and
MQCD do not agree, even qualitatively. The former has metastable vacua which the latter does not.
Open questions
• It would be nice to understand better the new M5 brane
mode which gives rise to the instability. Can we identify it
in the cigar CFT which describes the NS5 throat region?
• Do these conclusions still hold in the T-dual description,
where the brane configuration is replaced with
geometry? If so, what does this imply about “geometric
engineering”?
Open questions (cont’d)
• Can these conclusions be extended to the meta-stable
states of other SQCD-like theories? Are there patterns?
Are there general reasons why the meta-stable state of
SQCD has the wrong brane bending? (cf. the meta-stable state
of the KS geometry, which does not have problems with the bending…)
• Can we find any example where we have a controlled
description of a meta-stable state in both field theory and
string theory?