Meta-stable Vacua in SQCD and MQCD

David ShihHarvard 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 non-perturbative 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.

Visible sector (e.g. MSSM)

Messenger sector

HiddenSector (DSB)

Part of the complexity stems from the dearth of simple examples of DSB in field theory.

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.

String theory motivation

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:• Perturbative: • Exact:

hQi = h~Qi = 0hM i = (¤3N c ¡ N f mN f ¡ N c

0 )1=N c

• Beta function: Asymptotically free if Nf > 3Nc/2 (IR free if not)

• Superpotential:

• SUSY vacua:• Perturbative: no SUSY vacua!• Exact:

The “Magnetic” Theory

Let us understand this in more detail…

hM i = (¤3N c ¡ N f mN f ¡ N c0 )1=N c

“Rank condition” SUSY breaking

SUSY is broken at tree-level: (analogous to the O’Raifeartaigh model)

(rank Nf )(rank Nf -Nc )

(Of course, this SUSY breaking is a check of the duality. Otherwise, we would have had extra SUSY vacua.)

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:

Focus on the free-magnetic phase

Arbitrary matrix

Non-SUSY vacua of the magnetic theory

matrices

Classical vacua (up to global symmetries):

(Note: )

Potential for the pseudo-moduli

and parameterize a pseudo-moduli space.

This space is lifted in perturbation theory, since SUSY is broken.

and using the Coleman-Weinberg formula, we find

For , the magnetic theory has local SUSY-breaking vacua!

In fact, expanding around the point of maximal unbroken global symmetry:

Potential for the pseudo-moduli, cont’d

The mass-squareds are all positive!

Some properties of the SUSY vacua

• Global symmetries:

• Massless modes: – Goldstino– Goldstone bosons– gauginos and other fermions from unbroken discrete

• Vacuum energy:

What does this have to do with the “electric” SQCD theory?

Need Seiberg duality – “Electric” and “magnetic” theories become the same theory in the IR.

“Free magnetic phase”

.

.Magnetic theory is an IR free effective description of the electric theory at energies !

Electric

Magnetic

Seiberg Duality

• 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

Thus there is a meta-stable SUSY-breaking vacuum in SUSY QCD!

Meta-stable DSB in SQCD

Nc SUSY vacuaEffect of

Part II: Meta-stable Vacua in 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 (non-overlapping) field theory, IIA, or M-theory regime.

What is MQCD?

SQCD

MQCD MQCDMagnetic IIA brane configuration

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 meta-stable 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 , the NS5 branes bend due to their interactions with the D4 and D6 branes.

gs 6= 0

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

This configuration corresponds to origin of field space in the magnetic theory.

In the magnetic brane configuration, the analogue of the mass deformation appears to break supersymmetry.

Mass deformation (cont’d)

This provides a simple geometric realization of the SUSY-breaking vacuum of the magnetic theory. (Notice, however, that the NS’ bending is different now…)

The system can lower its energy by snapping together of the D4 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 non-perturbative 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

Holomorphic M5 branes

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

But then and the curve does not intersect the D6 brane!

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.)

x4+ix5 = m¡ cx6+cp (x6)2 ¡ 2cmx6+m2

1¡ c2

v(x6 =0) = m1¡ c 6=0

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 meta-stable 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?