## June 19, 2006

### Metastable Vacua

The number of vacua in an $\mathcal{N}=1$ supersymmetric field theory is, generically, dictated by the Witten index. This number is robust against perturbations, which makes dynamically breaking supersymmetry a challenge. One needs to start with a theory whose Witten index vanishes, and one needs to hope that the dynamical effect which break supersymmetry do not lead to runaway behaviour (like the runaway behaviour in supersymmetric QCD with $N_f \lt N_c$).

One of the nice insights of recent years is that we are not necessarily interested in true vacua (global minima of the effective potential). We might just as well be living in a metastable “false vacuum”, provided its lifetime is long enough. And, despite the paucity of true vacua (absent some symmetry), generic field theories can be chock-a-block with metastable vacua.

A while back, Intriligator, Seiberg, and Shih found a tractable example of metastable dynamical SUSY breaking in $\mathcal{N}=1$ SQCD with $N_c\lt N_f \lt \tfrac{3}{2}N_c$ flavours of massive quarks. This theory has $N_c$ supersymmetric vacua, at which the squark bilinear has an expectation value, $\langle \tilde{Q}_i Q_j\rangle \sim (\Lambda^{3N_c-N_f}\det m)^{1/N_c} (m^{-1})_{i j}$ where $m$ is the quark mass matrix (with eigenvalues $m_i$). But, for $m_i/\Lambda\ll 1$, the theory also has metastable nonsupersymmetric vacua whose properties can be reliably computed in the magnetic dual theory1, which is infrared-free for this range of $N_f$.

More recently, Ooguri and Ookouchi performed a similar analysis for the $\mathcal{N}=1$ $U(N_1)\times U(N_2)$ quiver gauge theory with bifundamental “quarks”, $Q,\tilde{Q}$, adjoint chiral multiplets, $X_1,X_2$ and a superpotential $W= W_1(X_1) + W_2(X_2) + \tr \tilde{Q} X_1 Q + \tr Q X_2 \tilde{Q}$ where the $W_i(X)$ are cubic. For $\tfrac{1}{2} N_1 \lt N_2 \lt \tfrac{2}{3} N_1$, this theory has an infrared-free dual description, as a $U(2N_2-N_1)\times U(N_2)$ gauge theory.

Again, they find supersymmetry-breaking metastable vacua near the origin in field space.

Seiberg duality, and the infrared-free dual description was a useful tool in analysing these models. But I think the general message is that supersymmetric gauge theories are chock-a-block with supersymmetry-breaking local minima and, with a suitable range of parameters, these metastable vacua can be very long-lived.

I like this for two reasons:

1. Because it shows that dynamical supersymmetry breaking is much “easier” than we thought. The models discussed above are easily embedded in string theory (more easily, one might say, than some of the more traditional examples of dynamical SUSY breaking).
2. Because (as sloganeered by Ooguri and Ookouchi) it shows that the existence of a “landscape” of metastable SUSY-breaking vacua is not some artifact of string theory, but is a generic feature of field theory.

1 $SU(N_f-N_c)$ SQCD, with $N_f$ dual “magnetic” quarks, a meson, $\Phi$, and a superpotential,

$W = \kappa \tilde{q} \Phi q + \tr(m \Phi)$
Posted by distler at June 19, 2006 11:09 AM

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### Re: Metastable Vacua

I don’t know whether to be happy or depressed that they find the field content of wrapped DBranes on compact Calabi Yaus are precisely the ones that ‘conspire’ to output supersymmetry breaking configuratiosn that are locally stable.

Worse, the ease of making the parametrizations for the decay rates small is troubling, and somewhat counterintuitive. One would have hoped for the existence of a selection method somewhere in there.

Posted by: Haelfix on June 19, 2006 6:17 PM | Permalink | Reply to this

### Re: Metastable Vacua

Thus, the “number of vacua” controlled by this index, is not expected to be connected to the number of massless particles we get in the theory?

Posted by: A Rivero on June 22, 2006 5:07 PM | Permalink | Reply to this

### Re: Metastable Vacua

The Witten index in the Intriligator et al model is $N_c$.

In the metastable vacua, there is always a massless fermion (the goldstino) and some number of massless goldstone bosons, corresponding to the broken flavour symmetry. All the other would-be massless modes are lifted by quantum corrections.

So, no, the Witten index has no relation to the number of massless particles in the metastable vacua.

Posted by: Jacques Distler on June 22, 2006 5:24 PM | Permalink | PGP Sig | Reply to this

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