The Witten’s paradox

Edward Witten is one of the greatest living physicists and also ranks high with mathematicians. He set the agenda for theoretical physics in several areas of research. He is mostly known for championing string theory and being one of few people that revolutionized the field. One of his major contributions to supersymmetry has been a deep understanding of its breaking. In a pair of famous papers (here and here) he put the foundations to our current understanding on the way supersymmetry can break and introduced the well-known Witten index. If a supersymmetric theory breaks supersymmetry then its Witten index is 0. This index is generally very difficult to compute and only perturbative or lattice computations can come to rescue. An important conclusion from Witten’s paper is that the well-known Wess-Zumino model in four dimensions does not break supersymmetry. Witten could rigorously justify this conclusion at small coupling but, at that time, an approach for strong coupling was missing and here Maldacena conjecture cannot help. Anyhow, he concluded that this should be true also for a strongly coupled Wess-Zumino model. Checks to this model in such a regime are rare. After I submitted a paper on arxiv last year (see here) I become aware of an attempt using Dyson-Schwinger equations that confirmed Witten conclusions for small coupling (see here). I have had an interesting mail exchange with one of the authors and this seems a promising approach, given authors’ truncation of Dyson-Schwinger hierarchy. Other approaches consider the Wess-Zumino model in two dimensions on the lattice. So, this appears a rather unexplored area , given the difficulties to cope with a strongly coupled theory, and Witten’s words appear like nails on a coffin to this theory.

I have worked out a lot of techniques to cope with strongly coupled theories and everywhere there is a perturbation going to infinity in a differential equation of any kind and so, I applied these ideas also to this famous model of supersymmetry. The idea is to prove that “supersymmetry has inside itself the seeds of its breaking“. The real issue at stake here is a correct understanding of the way supersymmetry breaks and recover in this way models that now appear to be defeated by data from LHC simply because the idea of symmetry breaking must be applied differently.

Of course, I do not aim to present a claim against the beautiful results given by Witten decades ago but just open up an interesting scientific question. So, considering that the Wess-Zumino model is just a theory of two scalar fields coupled to a Majorana spinor, its equations can be treated classically and so solved both for a strong and a weak coupling limit. I did this in a paper of mine (see here) and this paper has been accepted in these days in the Journal of Nonlinear Mathematical Physics as a letter. The classical solutions contradict the expectations giving a surviving of the supersymmetry at small coupling (as expected from Witten index for the quantum theory) while this does not happen for a strong coupling (formal limit of the coupling going to infinity). This is  a paradox, the Witten paradox, because classical solutions seem to break supersymmetry while the quantum theory does not.  So, we are left with a deep question: How is supersymmetry recovered by quantum corrections?

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

A. Bashir, & J. Lorenzo Diaz-Cruz (1999). A study of Schwinger-Dyson Equations for Yukawa and Wess-Zumino Models J.Phys.G25:1797-1805,1999 arXiv: hep-ph/9906360v1

Marco Frasca (2012). Classical solutions of a massless Wess-Zumino model arXiv arXiv: 1212.1822v2