# Dynamical supersymmetry breaking and Witten index

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Witten index, defined as ${\rm Tr}(-1)^F$, determines if supersymmetry is spontaneously broken or not for a given model. However, it is also known that supersymmetry can be dynamically broken. One could think of a mechanism (yet to be implemented if any) à la Nambu-Jona-Lasinio model for chiral symmetry in QCD. In this case, how does Witten index change? I think it should reflect the fact that, even if we are not requiring explicitly a selected ground state breaking symmetry, particle masses are lifted and a gap equation is satisfied.

A more general question is: if Witten index applies as well to dynamical breaking of supersymmetry, for all the models one can possibly conceive.

As usual, good references are welcome.

Thanks.

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user Jon

asked Nov 29, 2012
edited Dec 22, 2014

## 2 Answers

+ 4 like - 0 dislike

When ${\rm Tr}(-1)^F\neq 0$, then supersymmetry cannot be spontaneously broken. One could say that this basic fact is the very point of the Witten index.

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user Luboš Motl
answered Nov 29, 2012 by (10,278 points)
Yes, I know this. I have read Witten's paper. But how does this idea translate to dynamical breaking of symmetry? And if I would have a NJL mechanism?

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user Jon
+ 4 like - 0 dislike

Last time I checked, dynamical SUSY breaking was a particular kind of spontaneous SUSY breaking. It just means using certain non-perturbative effects to lift some of the flat directions in moduli space.

So Witten's index trick still works: You can prove that SUSY is unbroken by showing that the Witten index is non-zero.

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user user1504
answered Nov 29, 2012 by (1,110 points)
I think this is a right view.

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user Jon
It's the right answer, but I gave it -1 because it's exactly copied from my answer that was written 5 hours earlier. Why?

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user Luboš Motl
@LubošMotl: Jon didn't seem to realize that dynamical susy breaking is a form of spontaneous susy breaking and I thought that stating this explicitly was important. Your answer didn't directly address the point he was confused about. Anyways, no particular harm done. (And by the way, I'm the sole upvoter of your answer.)

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user user1504
It is unfortunate that this rather dippy question has 6 upvotes, while its two correct answers have (net) 1 each.

This post imported from StackExchange Physics at 2014-12-22 11:37 (UTC), posted by SE-user user1504

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