Renormalization group and dynamical symmetry breakdown in abelian gauge theories

Generalized renormalization group equations are used to analyze the dynamical mechanism of particle mass generation in terms of the Cornwall-Norton model both with and without cut-off. We look for solutions which contain non-zero physical masses of the two fermions (m₁, m₂) and of one of the vector bosons (μ) when the bare masses m_1Λ, m_2Λ, μ_Λ approach zero. For a theory without cut-off we obtain results which are similar to those of Cornwall. For a theory with cut-off the mass generation mechanism may only occur when a bare coupling constant α_Λ of the A_μ vector boson, which remains massless, exceeds some critical value α_c. In this case the fermion masses turn out to be of the superconductivity type.The model's “memory” of the nature of spontaneous symmetry breaking lim_{m1Λ, m2Λ → 0}m_1Λ/m_2Λ ≠ 1 is an indispensible factor for the vector boson to acquire a mass.     

Degrees of symmetry in quantum field theories

A hierarchy of possible symmetries in quantum field theory is defined, which reaches from a purely mathematical invariance to the conventional physical invariance, including the commonly discussed type of spontaneously broken symmetry (SBS). It is shown that one type of SBS, which is usually not considered, naturally leads to theories with an algebra of non-conserved currents and a non-linearly transforming phenomenological Lagrangian. An exactly solvable model is given and some general remarks are made.     

Absence of interaction in Lie field theories

It is shown that theories of relativistic Lie fields cannot lead to scattering or reactions, even if an infinite number of Lie fields is present.Supported in part by the National Science Foundation under Grant NSF GP 6036.     

Two-loop divergences of field theories with nonlinear symmetry

A general formula is obtained for two-loop counterterms of the field theories with the nonlinear realization of symmetry group.     

Cutoff dependence of vacuum properties for nucleon-meson quantum field theories

Pure vacuum effects for renormalizable nucleon-meson field theories are studied using a variant of the Pauli-Villars regularization scheme with finite cutoffs. In particular, the renormalized vacuum contribution to the one-nucleon-loop effective potential and scalar density are studied. It is found that one must use cutoffs of many GeV in order for these quantities to approximate well the infinite cutoff results and that a cutoff of two GeV or less yields the opposite sign from the infinite cutoff limit. This suggests that important contributions to these one-nucleon-loop vacuum quantities come from scales of order five GeV - a scale at which degrees of freedom other than nucleons and mesons may well be relevant.     

Problems with spontaneous breakdown in reggeon field theories

We prove a theorem which states that if a reggeon field theory undergoes spontaneous breakdown as a result of the pomeron intercept being above one, then the shifted Lagrangian must in general develop terms which are non-canonical from the point of view of reggeon Lagrangians. We discuss the interpretation of these terms and point out some of the problems associated with them.     

Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics

Two-dimensional lattice model is considered. The connected Lie groupG acts on a configuration space. The Gibbs potential assumed to be invariant under this action. We prove, that under general assumption on the potential, each Gibbs random field with this potential is alsoG-invariant.     

Hidden fermionic symmetry in conformal topological field theories

We point out that bosonic conformal coset modelsG _l ×G _k /G _l+k for all semi-simple Lie algebrasG have a hidden fermionic symmetry atl=0 (the central charge=0) and may be interpreted as twisted versions of some superconformal theories.     

Calculation of the regularized vacuum energy in cavity field theories

A novel technique based on Schwinger's proper time method is applied to the Casimir problem of the M.I.T. bag model. Calculations of the regularized vacuum energies of massless scalar and Dirac spinor fields confined to a static and spherical cavity are presented in a consistent manner. While our results agree partly with previous calculations based on asymptotic methods, the main advantage of our technique is that the numerical errors are under control. Interpreting the bag constant as a vacuum expectation value, we investigate potential cancellations of boundary divergences between the canonical energy and its bag constant counterpart in the fermionic case. It is found that such cancellations do not occur. Received: 19 March 1999 / Published online: 28 September 1999     

The general stationary gravitational vacuum field of cylindrical symmetry

The general stationary vacuum gravitational field of cylindrical symmetry as recently found by Davies and Caplan is even static. The possible Petrov types of the Riemann tensor areI,D orO. In spacelike infinity the spacetime becomes necessarily flat.     

HeiddenSL(n) symmetry in conformal field theories

We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of theSL(2, ) current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism. Thus there is aSL(2, ) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrainedSL(2, ) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of theSL(2, ) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of theSL(2, ) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of theSL(2, ) current algebra and its relation to theW _n -algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for theW _n -algebra despite its non-Lie-algebraic nature.This paper is dedicated to the memory of Vadik G. Knizhnik     

Hidden quantum group symmetry and integrable perturbations of conformal field theories

The hidden quantum group symmetry in the quantum Sine-Gordon model is found. This symmetry provides the possibility to restrict the operator algebra of the model to subalgebras. It is shown that these subalgebras are massive deformations of minimal conformal field theories.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Generalized local gauge symmetry and the Ward-Takahashi identities in unified field theories

We discuss the symmetry basis of unified field theories, i.e., the generalized concept of local gauge symmetry, and its physical implications. The generalized Ward-Takahashi identities and the explicit constraints among renormalization constants are derived by using the path integral in a specific model. These constraints are confirmed at the one-loop level.Work supported in part by the U.S. Energy Research and Development Administration.     

On vacuum energies and renomalizability in integrable quantum field theories

We compute for various perturbed conformal field theories the vacuum energies by means of the thermodynamic Bethe ansatz. Depending on the infrared and ultraviolet divergencies of the models, governed by the scaling dimensions of the underlying perturbed conformal field theory in the ultraviolet, the vacuum energies exhibit different types of characteristics. In particular, for the homogeneous sine-Gordon models we observe that once the conformal dimension of the perturbing scalar field is smaller or greater than 1/2, the vacuum energies are positive or negative, respectively. This behaviour indicates the need for additional ultraviolet counterterms in the latter case. At the transition points we obtain an infinite vacuum energy, which is partly explainable with the presence of several free fermions in the models studied.     

Absence of periodic solutions to scale invariant classical field theories

It is shown that in 3 + 1 Minkowski space classical field theories with energy-momentum θ_μν(x,t) satisfying θ₀₀(x,t) ? 0 and θ_μ^μ(x,t) = 0 have no finite energy periodic (or static) solutions. In particular this rules out classical glueball solutions. to Yang-Mills theories with compact gauge groups.     

Dynamics of vacuum excitations and quark confinement in quantum field theories

A unified approach to interacting vacuum excitations and quark confinement is formulated in quantum field theories with symmetry breakdown. Vacuum excitations are shown to be coherent clouds of Goldstone bosons or gauge bosons and are interpreted as new asymptotic extended particle states. They correspond to all dynamically possible space-time dependent Bose condensations of the Goldstone bosons in a given theory. Different configurations of vacuum excitations are connected to one another by a family of invariant boson transformations. As an example, the Nambu theory of interacting vortex strings is derived from a Nambu-Heisenberg quark-gluon field theory. The quarks can be completely confined to the strings while the gluons cluster in quantized magnetic flux bundles of penetration width m_v^?1 and provide a short range interaction force.     

QCD sum rules,the spontaneous breakdown of chiral symmetry and short-distance behaviour in lattice gauge theories

We analyze the behaviour that correlation functions ought to have on the lattice in order to reproduce QCD sum rules in the continuum limit. We formulate a set of relations between lattice correlation functions of meson operators at small time separation and the quark condensates responsible for spontaneous breakdown of chiral symmetery. We suggest that the degree to which such relations are satisfied will provide a set of consistency checks on the ability of lattice Monte Carlo simulations to reproduce the correct spontaneous chiral symmetry breaking of the continuum theory.     

Random source induced spontaneous symmetry breaking in scalar field theories

We study the effective potential for scalar field theories in the presence of gaussian random sources, coupled to the scalar field in a self-consistent way. We compute the effective potential both in the loop and in the 1/N expansions and find various instabilities. The only feasible instabilities are the ones induced by (formally) imaginary random sources. The pertinent phase transition is a first-order transition.     

Derivation and uniqueness of vacuum solutions of conformally covariant coupled nonlinear field equations

We derive the solutions of conformally covariant coupled Dirac and scalar fields including a nonlinear fermion self-coupling term for which the conformally covariant (not the canonical, nor the symmetric) energy-momentum tensor θ_μν vanishes. This “vacuum” state is degenerate.