Dynamical Symmetry Breaking and Mass Generation

We give a brief review of the physics of spontaneous symmetry breaking. The field-theoretic Goldstone theorem and its consequences are emphasized. In relativistic quantum theory, the emergence of particle masses is connected with spontaneous breaking of chiral symmetry. While in the Standard model of the electroweak interaction this is achieved by the Higgs mechanism, we show on a simple model that chiral symmetry may be broken dynamically by strong Yukawa interaction. The possibility of applying our results to electroweak interactions is discussed. Introduction Understanding the origin of particle masses is one of the most challenging unsolved problems in current high-energy physics. All known elementary interactions besides gravity are described by the Standard model, which acquired its up-to-date form in late sixties through the work of Weinberg [1967] and many others. The Standard model has passed successfully all experimental probes, to which it has so far been exposed. Yet, and despite the ingenuity of its creators, it suffers from a couple of diseases that bother the minds of the world’s most prominent theorists. The presence of particle masses in the Standard model Lagrangian is prohibited by the SU(2)L × U(1)Y gauge symmetry of the electroweak interaction. To overcome this difficulty, the Higgs mechanism has been suggested which leads to spontaneous breakdown of the electroweak symmetry by condensation of a scalar Higgs field. Although this approach is formally well defined and absolutely consistent, it is rather unsatisfactory physically as it is introduced phenomenologically, that is more or less by hand. It has been a dream of many particle physicists ever since the invention of the Standard model to find a more physical, dynamical explanation for electroweak symmetry breaking, and thus the origin of particle masses. All such attempts inevitably lead to introduction of new particles and strong forces. Many of them, the first perhaps being the technicolor theory [Weinberg, 1979], introduce new strong gauge interactions, taking advantage of a rather good knowledge of quantum chromodynamics. In this contribution we present a modest attempt in a different direction. We wish to show that chiral symmetry might be broken dynamically with just the Yukawa interaction, which is already present in the Standard model Lagrangian, provided it is strong enough. The plan of the paper is following. We start with an introductory review of the physics of spontaneous symmetry breaking, emphasizing its general features and wide applications. We explain the connection to the problem of fermion masses in relativistic quantum field theory. Next we work out a simple model with Abelian chiral symmetry in order to demonstrate dynamical mass generation by means of the Yukawa interaction. Starting from general considerations of symmetry and its realization in quantum theory, we end up with the Schwinger–Dyson equations whose non-perturbative solution yields the desired masses. The extension to the Standard model, which involves a non-Abelian gauged chiral symmetry and several fermion species, is discussed in the conclusions. Spontaneous symmetry breaking Symmetry plays a key role in analyzing most physical problems. It sometimes happens that the symmetry of the equations of motion is larger than the symmetry of their particular solution. In such a case we speak of spontaneous symmetry breaking (SSB). While in classical physics this seems rather trivial, in quantum field theory SSB has far-reaching consequences. Symmetry is introduced as an invariance of the Lagrangian or the action. By the Wigner theorem, it is implemented on the Hilbert space of states by unitary operators that commute with the Hamiltonian, leading to non-trivial degeneracy of the spectrum. On the other hand, the ground state is assumed to be non-degenerate, and it must therefore be invariant under the symmetry transformations. When SSB occurs, the ground state is by definition not invariant with respect to some of the symmetry operations. There are several examples of SSB both in non-relativistic and relativistic physics. The first class, which is probably more accessible to one’s intuition, includes for example the ferromagnet. WDS'05 Proceedings of Contributed Papers, Part III, 436–441, 2005. ISBN 80-86732-59-2 © MATFYZPRESS