I am a careful reader of scientific literature and an avid searcher for already published material in peer reviewed journals. Of course, arxiv is essential to accomplish this task and to satisfy my needs for reading. In these days, I am working on Dyson-Schwinger equations. I have written on this a paper (see here) a few years ago but this work is in strong need to be revised. Maybe, some of these days I will take the challenge. Googling around and looking for the Dyson-Schwinger equations applied to the well-known supersymmetric model due to Wess and Zumino, I have uncovered a very exciting track of research that uses Dyson-Schwinger equations to produce exact results in quantum field theory. The paper I have got was authored by Marc Bellon, Gustavo Lozano and Fidel Schaposnik and can be found here. These authors get the Dyson-Schwinger equations for the Wess-Zumino model at one loop and manage to compute the self-energies of the involved fields: A scalar, a fermion and an auxiliary bosonic field. Their equations are yielded for three different self-energies, different for each field. Self-energies are essential in quantum field theory as they introduce corrections to masses in a propagator and so enters into the physical part of an object that is not an observable.

Now, if you are in a symmetric theory like the Wess-Zumino model, such a symmetry, if it is not broken, will yield equal masses to all the components of the multiplet entering into the theory. This means that if you start with the assumption that in this case all the self-energies are equal, you are doing a consistent approximation. This is what Bellon, Lozano and Schaposnik just did. They assumed from the start that all the self-energies are equal for the Dyson Schwinger equations they get and go on with their computations. This choice leaves an open question: What if do I choose different self-energies from the start? Will the Dyson-Schwiner equations drive the solution toward the symmetric one?

This question is really interesting as the model considered is not exactly the one that Witten analysed in his famous paper on 1982 on breaking of a supersymmetry (you can download his paper here). Supersymmetric model generates non-linear terms and could be amenable to spontaneous symmetry breaking, provided the Witten index has the proper values. The question I asked is strongly related to the idea of a supersymmetry breaking at the bootstrap: Supersymmetry is responsible for its breaking.

So, I managed to numerically solve Dyson-Schwinger equations for the Wess-Zumino model as yielded by Bellon, Lozano and Schaposnik and presented the results in a paper (see here). If you solve them assuming from the start all the self-energies are equal you get the following figure for coupling running from 0.25 to 100 (weak to strong):

It does not matter the way you modify your parameters in the Dyson-Schwinger equations. Choosing them all equal from the start makes them equal forever. This is a consistent choice and this solution exists. But now, try to choose all different self-energies. You will get the following figure for the same couplings:

This is really nice. You see that exist also solutions with all different self-energies and supersymmetry may be broken in this model. This kind of solutions has been missed by the authors. What one can see here is that supersymmetry is preserved for small couplings, even if we started with all different self-energies, but is broken as the coupling becomes stronger. This result is really striking and unexpected. It is in agreement with the results presented here.

I hope to extend this analysis to more mundane theories to analyse behaviours that are currently discussed in literature but never checked for. For these aims there are very powerful tools developed for Mathematica by Markus Huber, Jens Braun and Mario Mitter to get and numerically solve Dyson-Schwinger equations: DoFun anc CrasyDSE (thanks to Markus Huber for help). I suggest to play with them for numerical explorations.

Marc Bellon, Gustavo S. Lozano, & Fidel A. Schaposnik (2007). Higher loop renormalization of a supersymmetric field theory Phys.Lett.B650:293-297,2007 arXiv: hep-th/0703185v1

Edward Witten (1982). Constraints on Supersymmetry Breaking Nuclear Physics B, 202, 253-316 DOI: 10.1016/0550-3213(82)90071-2

Marco Frasca (2013). Numerical study of the Dyson-Schwinger equations for the Wess-Zumino

model arXiv arXiv: 1311.7376v1

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

Markus Q. Huber, & Jens Braun (2011). Algorithmic derivation of functional renormalization group equations and

Dyson-Schwinger equations Computer Physics Communications, 183 (6), 1290-1320 arXiv: 1102.5307v2

Markus Q. Huber, & Mario Mitter (2011). CrasyDSE: A framework for solving Dyson-Schwinger equations arXiv arXiv: 1112.5622v2