Spontaneous symmetry breaking and constant gauge transformations

Symmetry breaking solutions of several model theories are investigated with the result that constant gauge transformations of the fields describing zero mass Goldstone particles are responsible for the formal possibility of the spontaneous symmetry breaking.

     

Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism

Spontaneous symmetry breakings in indefinite metric quantum field theories are analyzed and a generalization of the Goldstone theorem is proved. The case of local gauge quantum field theories is discussed in detail and a characterization is given of the occurrence of the Higgs mechanism versus the Goldstone mechanism. The Higgs phenomenon is explained on general grounds without the introduction of the so-called Higgs fields. The basic property is the relation between the local internal symmetry group and the local group of gauge transformations of the second kind. Spontaneous symmetry breaking ofc-number gauge transformations of the second kind is shown to always occur if there are charged local fields. The implications about the absence of mass gap in the Wightman functions and the occurrence of massless particles associated with the unbroken generators in the Higgs phenomenon are discussed.     

Spontaneous dynamical breaking of gauge symmetry in dual models

The consequences of gauge invariance are re-examined in dual models with unit intercept when the dimension of space-time has the maximum value compatible with the absence of ghosts. In that case, reggeon-reggeon bound states of zero mass are formed already at second order of perturbation theory. Gauge invariance no longer guarantees that the initially massless SU(3) singlet vector meson remains massless, and the mixing between the reggeon and pomeron sectors of the model yields a massive unitary singlet vector meson already at order g². A Lagrangian model which exhibits similar features to the dual situation is discussed.     

Spontaneous symmetry breaking in annealed and quenched gauge field models

The structure of annealed and quenched models with localU(1) gauge invariance is studied in terms of the Helmholtz free energy. The first non-trivial, or one-loop, account of fluctuations in the annealed model suggests that spontaneous symmetry breaking occurs in two and three dimensions, through a first-order phase transition. Within the same approximation scheme, the quenched model displays a continuous phase transition. A more complete account of the fluctuations in the annealed model changes the nature of the transition to a continuous one, whereas spontaneous symmetry breaking is then absent with quenched disorder.     

Spontaneous symmetry breaking in annealed and quenched gauge field models

The structure of annealed and quenched models with localU(1) gauge invariance is studied in terms of the Helmholtz free energy. The first non-trivial, or one-loop, account of fluctuations in the annealed model suggests that spontaneous symmetry breaking occurs in two and three dimensions, through a first-order phase transition. Within the same approximation scheme, the quenched model displays a continuous phase transition. A more complete account of the fluctuations in the annealed model changes the nature of the transition to a continuous one, whereas spontaneous symmetry breaking is then absent with quenched disorder.     

Spontaneous symmetry breaking in Reggeon field theory

We consider a Reggeon field theory when the bare or input Regge intercept α_O is greater than one. This corresponds to a negative mass squared term in conventional field theory and allows for a spontaneous symmetry break-down. A theory with Regge intercept at one emerges, restoring the Froissart bound by t-channel considerations alone. In our elementary example the resulting bare trajectory is nearly of the square root variety familiar from s-channel eikonalization of models which violate the Froissart bound.     

Proof of chiral symmetry breaking in lattice gauge theory

Using the method of infrared bounds and partial-integration formulas, we prove that there is a chiral phase transition in four-dimensional strongly coupled lattice gauge theory with gauge group U(N) and staggered fermions for all N5.     

Spontaneous symmetry breaking of affine field theory

It is possible to construct non-Abelian field theories by gauging Kac-Moody algebras. Here we discuss the spontaneous symmetry breaking of such theories via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra.     

Spontaneous symmetry breaking of Slavnov symmetry A restriction on the gauge condition

It is shown that the condition of vanishing vacuum expectation value of the gauge operator, using a gauge-fixing term quadratic in this operator, does not necessarily follow from the Slavnov identity, due to a possible spontaneous breakdown of the Slavnov symmetry. For a consistent renormalization such a condition may have to be imposed order by order in perturbation theory, depending on the choice of the gauge. This restriction on non-singular gauges is particularly relevant for the discussion of the spontaneously broken realizations of the gauge symmetry.     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.     

Spontaneous symmetry breaking in massless field theories: Dimensional transmutation and gauge independence

In a reformulation of the Coleman and Weinberg program, we study the parametrization of the spontaneous symmetry breaking solutions of massless field theories. Our analysis is based on a full use of the concept of the dimensional transmutation, namely the parametrization in terms of the independent invariants of the underlying renormalization group. This invariant parametrization provides a convenient setting for the operator interpretation in a given Fock space and is shown to be a powerful tool in studying general aspects of the theory beyond the one-loop approximation. In particular, the gauge independence of the physical matrix elements in the spontaneously broken solution of massless scalar electrodynamics is demonstrated to all orders in perturbation theory.     

Proof of chiral symmetry breaking in strongly coupled lattice gauge theory

We study chiral symmetry in the strong coupling limit of lattice gauge theory with staggered fermions and show rigorously that chiral symmetry is broken spontaneously in massless QED and the gauge-invariant Nambu-Jona-Lasinio model if the dimension of spacetime is at least four. The results for the chiral condensate as a function of the mass imply that the mean-field approximation is an upper bound for this observable which becomes exact as the dimension goes to infinity. For the model with gauge groupU(N),N=2,3,4, we prove that chiral long-range order exists at zero mass in four or more dimensions. Address after August 1991: Mathematics Department, University of British Columbia, Vancouver, Canada V6T1Y4     

Chiral symmetry breaking in the action formulation of lattice gauge theory

We show, in the euclidean path-integral formulation of strong-coupling lattice gauge theory, that continuous chiral symmetry is dynamically broken, and obtain the standard current algebra result that $\text{m}{}_{\mathrm{<mtext>pseudo-Goldstone</mtext>}}{}^2\text{～ m}{}_{\mathrm{<mtext>quark</mtext>}}\text{〈}\text{ψ}\text{ψ〉}$. We also remark that the center of the gauge group does not seem very relevant for this result; chiral symmetry breaking is a property of strong-coupling lattice theories both in the case where quark color is confined, and also in the case where it is screened by gauge field fluctuations.     

Dynamical symmetry breaking in the SU (2) gauge theory

Properties of vacua of the SU(2) gauge theory containing massless and massive fermions are investigated within the one-loop approximation. As a result of a condensation of composite scalar made of gauge fields, some gauge fields acquire a mass and the original SU(2) gauge symmetry is suggested to breake down to U (1). We evaluate the effective potential for the constant magnetic field and then extract the so-called dielectric permeability κ from it. The phase is called paramagnetism for positive κ, perfect paramagnetism for vanishing κ and ferromagnetism for negative κ. The choice of a favorable phase is determined by the value of the coupling constant and the number and the mass of fermions. The perfect paramagnetic phase is most precisely studied. It is shown that solitons with non-zero charges carry divergent energies. Then, the electric flux around the charge is shown to be squeezed into a string in that phase.     

Composite operator calculation of chiral symmetry breaking in color gauge theory

In the framework of the formalism of Cornwall et al. for composite operators, we develop a new method for the calculation of the effective potential. We apply the method to the study of chiral symmetry breaking in an SU(N) color gauge theory with n massless flavors and derive analytically the complete effective potential at two fermion loops. We find that in this approximation the theory has two phases: the symmetric phase and the broken phase into the diagonal flavor sub-group.     

Spontaneous chiral symmetry breaking for a U(N) gauge theory on a lattice

We study the breakdown of chiral invariance by calculating, in the infinite coupling, large-N limit, the generating functional of a U(N) gauge theory with one fermion, expressed on a lattice with the naive, chiral symmetric action. We compute the link integral over the gauge fields and the expression obtained after the integration over the fermions is recast under the form of a generating functional for bosonic fields. Then, a saddle-point method allows the calculation of the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ for which a non-zero value signals the spontaneous breakdown of chiral symmetry. The analysis of the fluctuations around the saddle point allows one to exhibit the Goldstone modes corresponding to those global symmetries of the fermionic lattice action which are simultaneously broken.