Using the effective potential for composite operators but without using a variational approach, we show the possible existence of a dynamical phase transition from a massive phase to a massless one. The problem of the spontaneous breakdown of chiral symmetry with a composite operator in non soluble theories such as four-dimensional QED is an old and open one. ' Some time ago we showed, using the formalism of the effective potential for composite operators (EPCO), that in the ladder

more »

... , in quantum electrodynamics in a continuum space-time with a massless fermion, the latter gets a mass for every nonzero coupling constant. In this case the Goldstone theorem is evaded. The picture that arises, taking our results seriously, is in close analogy with the BCS theory of superconductivity in which any charge different from zero produces a condensate if not thermally disrupted. In Ref. 3 we calculated the EPCO using a variational approach and two asymptotic solutions for the Schwinger-Dyson (SD) equation for the fermion propagator (see below) which are valid for 0&a=e /4~&m. /3; that is, we are able to search for a critical coupling constant in the above interval for a. On the other hand, it is possible to consider the mechanism of spontaneous chiral-symmetry breaking in QED based on the analogy between this phenomenon and that of the "fall" of a particle to the center. In this formalism the critical coupling constant is a, =m/3, which separates the massless (a & a, ) and the massive (a & a, ) phases. A third approach to the problem of chiral phase transition is the lattice technique. In noncompact lattice QED in the quenched approximation the critical coupling constant a, =3.75 has been found. However, using an improved action (having the same infrared behavior but differing in the ultraviolet part) the critical constant becomes a, =2.00. The authors of Ref. 7 have interpreted these results as an indication that the critical coupling constant is nonuniversal and is strongly influenced by the short-distance behavior of the gauge field theory. When fermions are introduced in the theory the phase transition survives. But in Ref. 8 fermions have a relatively large bare mass: m =0.25 and 0.20 in lattice units. Recently a study of the chiral phase transition in compact QED with light fermions (m =0. 1,0.2) gave a critical coupling constant lower than the previous ones: a, =1.119 for m =0.10 and a, =1.088 for m =0.25. In Ref. 9, the existence of a first-order transition for m =0. 10 is also suggested. Recently a lot of new and interesting results have been obtained in noncompact QED. For example, with four species of light fermion (i.e. , Nf =4) and bare fermion mass m =0.0125 in lattice units simulation data confirm that there is a second-order phase transition to a strongcoupling phase where chiral symmetry is broken. ' There is also a finite-size effect which is a nontrivial effect which can be interpreted as an indication that the phase transition is associated with an ultraviolet-stable fixed point of an interaction theory of meson bound states. The theory also shows a real temperature and flavor number dependence. " We can think that our results using EPCO with a variational approach which gives us a, =0 (Ref. 3) could be consistent with the collapse picture and lattice technique " in which a, 1 if there is a phase transition with a critical coupling constant a, ', with 0(a, ' &1 but now the transition being from a massive phase (a & a, ' ) to a massless one (a & a, '). Then the other phase transition with a,~1 from a massless phase to a massive one could occur. If this is true the reason we did not find the critical coupling constant a, ' in Ref. 3 could be an artifact of the variational approach or of the two-loop effective potential. On the other hand, in order to have physically significant results we must use in the effective potential exact solutions of the SD equation or a variational approach. Of course, exact solutions to the nonlinear SD equation are not known; for this reason the variational approach is the preferred one in the literature for both Abelian ' ' and non-Abelian' theories. In this paper we want to point out that in the case of QED in the latter approximation and with -p~~w e have asymptotic solutions and for this reason we can use them with some confidence in the effective potential without using a variational approach. This could not be done in QCD in which even asymptotic solutions are only approximates. ' ' %'e shall use the usual situation ' ' of the free vertex and 42 719