WDS'05 Proceedings of Contributed Papers, Part III, 436–441, 2005. ISBN 80-86732-59-2 © MATFYZPRESS Dynamical Symmetry Breaking and Mass Generation T. Brauner and J. Hoˇsek Nuclear Physics Institute, AS CR, Reˇz,ˇ Czech Republic Abstract. 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. 436 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION There, the Hamiltonian is invariant under rotations of electron spins. In the ground state, however, the spins of all electrons are aligned, breaking the full rotation SO(3) group down to the SO(2) subgroup of rotations about the axis of spontaneous magnetization, which constitutes the order parameter for the phase transition to the ferromagnetic state. The direction of the magnetization is arbitrary and is induced by some random external perturbation that slightly breaks the rotation symmetry. Other non-relativistic examples are the superconductor, where the broken symmetry is the electro- magnetic gauge invariance, or even any crystallic solid, where full translation symmetry is broken by the crystal lattice down to some discrete subgroup. Relativistic, especially particle physics offers a variety of examples of SSB, e.g. the gauge SU(2)L U(1)Y invariance of the electroweak interaction and its exten- × sions to larger symmetry groups in models of grand unification, or the color SU(3)c in superconducting phases of quantum chromodynamics. The experience gained from the above listed examples may be summarized in the observation that SSB is always characterized by the existence of an order parameter. This can acquire various values, which are formally connected by the broken symmetry transformations. There are therefore several degenerate ground states, each specified by the particular value of the order parameter. This is the intuitive picture of SSB. Formally, however, the broken symmetry is not implemented on the Hilbert space by unitary operators and thus is not manifested in the degeneracy of the spectrum. As a consequence, different ground states are not connected by a unitary transformation. They are unitarily inequivalent and the Hilbert spaces built above them are completely orthogonal — they represent different worlds with different values of the order parameter. Chiral symmetry We shall now in more detail comment on the connection of SSB and fermion mass generation. The 1 Lagrangian of a free relativistic spin- 2 (Dirac) fermion contains two separate pieces for the left- and right- handed chiral components of the Dirac field, 0 = ψ¯Ri∂ψ/ R + ψ¯Li∂ψ/ L. It is invariant under independent L phase transformations of the left and right fields, that is the Abelian chiral group U(1)R U(1)L. × Now the Dirac mass term, δ m = m(ψ¯RψL + ψ¯LψR), intertwines the left and right fields so that they can no more be rephased independently.L − The mass term is invariant only under the diagonal vector U(1)V . It follows that non-zero Dirac masses can arise in a chirally-invariant theory only after the chiral symmetry has been spontaneously broken. The mass m itself is the order parameter here. The ground state is chosen so that m is real and positive and corresponds directly to the physical mass of the particle. Goldstone theorem The Goldstone theorem [Goldstone et al., 1962] is one of a few exact and general results in quantum field theory, and is the starting point for any consideration about broken symmetry. It states that whenever a continuous symmetry is spontaneously broken, there is an excitation branch in the spectrum of the theory (the Nambu–Goldstone mode) for which E(k) 0 as k 0. For Lorentz-invariant theories, this statement considera→bly simplifies:→ the Nambu–Goldstone (NG) mode is a spinless particle with zero mass. It also holds that the number of NG bosons is equal to the number of broken symmetry generators. For non-relativistic systems the NG boson counting is more complicated [Nielsen and Chadha, 1976]. It turns out that the number of NG bosons may be smaller than the number of broken generators provided some of the symmetry generators develops non-zero density in the ground state. This is the case for the ferromagnet mentioned before — there are two broken generators, but a single NG mode, the magnon. Before concluding the review part of the contribution we note that there are also symmetries that are not exact, e.g. the chiral SU(3)R SU(3)L symmetry of quantum chromodynamics, which is slightly broken by u, d, and s quark masses.× When such a symmetry is spontaneously broken, one naturally expects a nearly massless particle (the pseudo-Nambu–Goldstone boson) to appear. However, no theorem analogous to Goldstone’s exists for approximate symmetries. Abelian model of chiral symmetry breaking Now we shall introduce our model for dynamical breaking of chiral symmetry. We shall give a brief description and summarize the results, full details may be found in our recent paper [Brauner and Hoˇsek, 2005]. Trying to keep the idea as simple as possible, we consider the theory of a complex scalar field φ with mass M and two massless Dirac fermions ψ1,2, coupled by the Yukawa interaction Yukawa = y1 ψ¯1Lψ1Rφ + ψ¯1Rψ1Lφ† + y2 ψ¯2Rψ2Lφ + ψ¯2Lψ2Rφ† . (1) L 437 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION A curious reader might wonder why we include two different fermion species. The reason is that with just one fermion our model would suffer from an axial anomaly. This is no problem as long as the anomalous symmetry is global, as it actually is in our case. However, as we already mentioned above, our ultimate goal is to apply the same idea to electroweak symmetry breaking. For a gauge invariance the anomaly would be a disaster, and we therefore choose to remove it from the very beginning.