CERN-TH/96-195 Strongly Coupled QED ∗

A short review of some of the most relevant contributions to non-perturbative QED is done. Since a Gaussian behaviour of QEDà la λφ 4 has been ruled out by the numerical data, I analyse the other two most reliable scenarios, i.e. trivialityà la Nambu-Jona Lasinio and non-Gaussian critical behaviour. I give a suggestive theoretical argument against a Gaussian behaviour of QEDà la Nambu-Jona Lasinio, and show how the numerical data for the susceptibility at the critical point of QED support this result. 1 1. Introduction Quantum Electro-Dynamics (QED) is the most standard and popular example of a non-asymptotically free gauge theory. Due to the peculiar form of its β renormalization group function, as obtained in perturbation theory, this model suffers from the Landau pole problem, which means that the renormalized coupling increases with the energy and diverges at some finite energy scale. The only way to construct a consistent theory in the infinite cutoff limit is to put the renormalized coupling equal to zero and the model is therefore trivial. In spite of that, QED is also the most successful quantum field theory from a phenomenological point of view. In fact, at energy scales much lower than the cutoff scale, QED describes perfectly the electromagnetic interactions. There is no doubt that QED incorporates very well the main features of electromagnetic interactions between electrons and photons in Nature. There are several possible issues to the triviality problem of perturbative QED: i. Non-perturbative contributions to the β function, which would be negligible at small energies, become relevant at large energy scales and change the structure of the β function. An ultraviolet stable or non-Gaussian fixed point could appear in this way as driving to a non-trivial interacting theory in the infinite cutoff limit. This issue includes also the possibility that a summation of all the perturbative series be able to change the structure of the β function. Therefore " non-perturbative contributions " should be understood in a wide sense. ii. At large energy scales, and in the context of Unification Models for gauge interactions, other physical interactions between particles become relevant ; these new interactions are responsible for the necessary changes in the β function in order to get a non-trivial model. iii. The real world is only described by effective theories, which are interacting at small energy scales compared with the cutoff scale but are trivial if the cutoff is sent …