Long range dynamics and broken symmetries in gauge models. The Stückelberg-Kibble model

The general algebraic features associated to long range dynamics like the problem of removing the infrared cutoff, the definition of the algebraic dynamics and the occurrence of variables at infinity, the essential localization (seizing of the vacuum), the effective dynamics and its covariance group (dynamical symmetry group), the generalization of Goldstone's theorem and the non-trivial Goldstone spectrum, the mass/energy gap generation by the non-trivial classical motion of the variables at infinity are explicitly shown in the Kibble model as a prototype of gauge models exhibiting the Higgs phenomenon. The relation between mass generation in the Higgs phenomenon and the plasma energy gap is also discussed.Work supported in part by INFN, Sezione di Pisa     

Vacuum instabilities in models with spontaneous symmetry breaking

Adopting as a reference a simple model with spontaneously broken symmetry we show that the extra massless field present in the three approximation in addition to the true Goldstone bosons may induce, through the radiative corrections to its vacuum expectation value, infrared effects which are not compensable without spoiling the symmetry itself. We further extend the analysis to generic lagrangian field models with spontaneous symmetry breaking and prove that the only constraint to their renormalizability arises from the radiative corrections to the vacuum expectation value of the massless fields, except for the true Goldstone bosons which never induce such pathologies.     

On free massless (pseudo)scalar quantum field theory in 1+1-dimensional space-time

We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that in such a quantum field theory (i) a continuous symmetry of (pseudo)scalar field translations is spontaneously broken, (ii) Goldstone bosons appear as quanta of a free massless (pseudo)scalar field and (iii) there is a non-vanishing spontaneous {magnetization}. In spite of the existence of a spontaneous {magnetization} the main inequality between vacuum expectation values of certain operators which have been used for the derivation of the Mermin–Wagner–Hohenberg theorem (C. Itzykson and J.-M. Drouffe, {Statistical field theory}, Vol. I, 1989, pp. 219–224) is fulfilled. Received: 19 December 2001 / Revised version: 31 March 2002 / Published online: 14 June 2002     

On how to count Goldstone bosons

A generalization of Goldstone's theorem is presented which is valid for theories either with or without relativistic invariance. The central suggestion is that, under certain specified assumptions, all Goldstone bosons can be divided into two classes, termed type I and type II, in accordance with the behaviour of their dispersion laws. A Goldstone boson is a member of either the first or the second class according as its energy, in the limit of long wavelengths, is proportional to an odd or an even power of its momentum, respectively. The major result then is that, if each Goldstone boson of type I is counted once and that of type II is counted twice, the total number of “bosons” so obtained is always equal to or greater than the number of symmetry generators that are spontaneously broken. An immediate corollary is the familiar result that for relativistically invariant theories the number of Goldstone bosons can never be less than the number of spontaneously broken generators. Throughout the proof of the above result particular emphasis is placed on theories which are not Lorentz invariant.     

Goldstone fermion couplings and supersymmetry breaking in superstring models

We re-examine supersymmetry breaking in the observable sectors of superstring-inspired supergravity models by computing Goldstone fermion couplings at the one-loop level. We find that a single global U (1) phase invariance is sufficient to forbid masses for gauge non-singlet chiral scalar bosons, and that Heisenberg symmetry is not necessary.     

Spontaneous breaking of the Galilei group and the plasmon energy gap

The generalization of Goldstone's theorem to dynamics with long range interactions is applied to the Coulomb Fermi gas in uniform charge background. It is shown that the plasmon energy spectrum for can be exactly derived from the spontaneous breaking of the Galilei boosts, with the plasmon excitations playing the rôle of Goldstone bosons. The connection with the appearence of variables at infinity in the dynamics of local variables and with a “classical dynamics at infinity” is discussed.     

UL(N) x UR(N)-Invariant Four-Fermion Interactions and Nambu-Goldstone Mechanism at Finite Temperature

In a chiral U_L(N) x U_R(N) fermion model of NJL-form, we prove that, if all the fermions are assumed to have equal masses and equal chemical potentials, then at the finite temperature T below the symmetry restoration temperature T_c, there will be N² massive scalar composite particles and N² massless pseudoscalar composite particles (Nambu-Goldstone bosons). This shows that the Goldstone theorem at finite temperature for spontaneous symmetry breaking U_L(N) x U_R(N) → U_L+R(N) is valid and consistent with the real-time formalism of thermal field theory in this model.     

Spontaneous breakdown in two dimensional space-time

We prove that in two-dimensional space-time, symmetry transformations which are generated by Poincaré covariant currents can not be spontaneously broken. This is also the case with the dilation current. We argue that other currents which involve explicit space-time dependence might lead to spontaneously broken symmetries accompanied by massless Goldstone bosons. We construct a trivial example where this phenomenon occurs.     

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.     

A diquark model for baryons containing one heavy quark

We present a phenomenological ansatz for coupling a heavy quark with two light quarks to form a heavy baryon. The heavy quark is treated in the heavy mass limit, and the light quark dynamics is approximated by propagating scalar and axial vector ‘diquarks’. The resulting effective lagrangian, which incorporates heavy quark and chiral symmetry, describes interactions of heavy baryons with Goldstone bosons in the low energy region. As an application, the Isgur-Wise form factors are estimated.     

A diquark model for baryons containing one heavy quark

We present a phenomenological ansatz for coupling a heavy quark with two light quarks to form a heavy baryon. The heavy quark is treated in the heavy mass limit, and the light quark dynamics is approximated by propagating scalar and axial vector ??diquarks??. The resulting effective lagrangian, which incorporates heavy quark and chiral symmetry, describes interactions of heavy baryons with Goldstone bosons in the low energy region. As an application, the Isgur-Wise form factors are estimated.     

Chiral crossover characterized by Mott transition at finite temperature

We discuss the proper definition for the chiral crossover at finite temperature,based on Goldstone's theorem.Different from the commonly used maximum change in chiral condensate,we propose defining the crossover temperature using the Mott transition of pseudo-Goldstone bosons,which,by definition,guarantees Goldstone's theorem.We analytically and numerically demonstrate this property in the frame of a Pauli-Villars regularized NJL model.In an external magnetic field,we find that the Mott transition temperature shows an inverse magnetic catalysis effect.     

Kaon condensation in the color–flavor-locked phase of quark matter, the Goldstone theorem, and the 2PI Hartree approximation

At very high densities, QCD is in the color–flavor-locked phase, which is a color-superconducting phase. The diquark condensates break chiral symmetry in the same way as it is broken in vacuum QCD and gives rise to an octet of pseudo-Goldstone bosons and a superfluid mode. The lightest of these are the charged and neutral kaons. For energies below the superconducting gap, the kaons are described by an O(2)×O(2)-symmetric effective scalar field theory with chemical potentials. We use this effective theory to study Bose-condensation of kaons and their properties as functions of the temperature and the chemical potentials. We use the 2-particle irreducible effective action formalism in the Hartree approximation. The renormalization of the gap equations and the effective potential is studied in detail and we show that the counterterms are independent of temperature and chemical potentials. We determine the phase diagram and the medium-dependent quasiparticle masses. It is shown that the Goldstone theorem is satisfied to a very good approximation. The effects of imposing electric charge neutrality is examined as well.     

Finite size effects and spontaneously broken symmetries: the case of theO(4) model

Finite volume numerical simulations of scalar models with continuous symmetry face strong finite size effects in the broken phase due to the presence of light Goldstone states. In the region where the light Goldstone bosons dominate the dynamics of the system universal finite size scaling formulae are predicted by chiral perturbation theory. Introducing a finite external source one can determine infinite volume, zero external source physical quantities from finite volume observables. Here we apply this theoretically controlled approach to the 4 dimensionalO(4) scalar model. All of our numerical results are in excellent agreement with the predicted finite size scaling forms. We confirm earlier results at zero external source where the infinite volume limit was approximated by projecting the fields to the direction of the magnetization.     

Self-dual tensors and gravitational anomalies in 4n + 2 dimensions

Starting from a manifestly Lorentz- and diffeomorphism-invariant classical action we perform a perturbative derivation of the gravitational anomalies for chiral bosons in 4n + 2 dimensions. The manifest classical invariance is achieved using a newly developed method based on a scalar auxiliary field and two new bosonic local symmetries. The resulting anomalies coincide with the ones predicted by the index theorem. In the two-dimensional case, moreover, we perform an exact covariant computation of the effective action for a chiral boson (a scalar) which is seen to coincide with the effective action for a two-dimensional complex Weyl fermion. All these results support the quantum reliability of the new, at the classical level manifestly invariant, method.     

Internal symmetry currents and supersymmetric effective lagrangians

A supersymmetric and non-linearly realized internally symmetric action is constructed from the super Kähler potential of Goldstone scalar superfields. Noether's theorem in superspace is derived and the associated superfield of currents defined. The currents are used to derive a super Dashen formula relating the (quasi) Goldstone masses and decay constants to the symmetry breaking part of the theory and to supersymmetrically gauge the invariant subgroup as well as the full group.     

在高密度下核物质的色超导性

建立了相变热力学理论和场论的关系. 强调在量子场论中必须引进序参量场， 则相变的讨论就类似于Goldstone bosons 的产生. 如果只讨论一级相变， Goldstone bosons场就足够了; 如果要讨论二级相变， 则必须讨论一系列的场， 这些场构成一个对称群的表示. 另外, 也将这一思想用到色超导中. In this paper we built a relation between the thermodynamical theory of the phase transition and field theory. We emphasized that in the quantum field theory we have to introduce the order parameter fields. Then the discussion of the phase transition is closed to the creation of the Goldstone bosons. If we only discuss the first order transition, the Goldstone bosons fields are sufficient. If we want to discuss the second order transition, we have to discuss a set of fields that constructs a representation of a symmetry group. We also apply this concept to color superconductivity.     

Electroweak interactions and high-energy limit

An introduction to the equivalence theorem for longitudinally polarized vector bosons and unphysical Goldstone scalars in electroweak theories is presented and the problem of high-energy behaviour of scattering amplitudes in the Standard Model is briefly reviewed. The equivalence theorem is first illustrated on simple examples, and the idea of a general proof based on the concept of BRS symmetry is then explained. To make the treatment self-contained, basics of the Standard Model are summarized in the appendix.     

On vector Goldstone boson

The possibility that higher dimensional field theories are broken spontaneously, through the usual Nambu-Goldstone mechanism, to 4-dimension is explored. As a consequence, vector Goldstone bosons can arise in this breaking of Lorentzian symmetry from higher dimension to 4-dimension. This can provide a simple mechanism for reduction to 4-dimension in theories with extra dimensions. Received: 3 January 2002 / Published online: 14 March 2003     

τ→ν3π and the A1 resonance

We consider the possibility that the family group may be a spontaneously broken continuous global symmetry. In the context of grand unification, the couplings of the associated Goldstone bosons to fermions can be sufficiently suppressed so as to satisfy the phenomenological bounds. For a maximal family symmetry this requires a large number of Higgs fields.