Continuum limit of A Z2 lattice gauge theory

Phase diagrams of lattice gauge theories have in several cases lines of first-order transitions ending at points at which continuous (second-order) transitions take place. In the vicinity of this critical point, a continuum field theory may be defined. We have analyzed here a Z₂ gauge plus matter model (which has no formal continuum limit) and identified the critical point with a usual Ø⁴, globally Z₂ invariant, field theory. The analysis relies on a mean field functional formalism and on a loop-wise expansion around it, which is reviewed.     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

Kac-Moody symmetries of critical ground states

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su(2)_k=1, su(3)_k=1, and the su(4)_k=1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.     

12C(e,e′ p) missing momentum distributions in HF-Sk3 and RPA-Sk3 continuum theories

We study the momentum distribution or reduced cross-section for electron induced 1p _3/2 proton knockout from¹²C in parallel kinematics. We refer to continuum self-consistent HF-Sk 3 and RPA-Sk 3 theories with a full treatment of the one-nucleon energy continuum. The PWIA limit is also shown. The¹²C(e, e′p ₀) missing momentum distribution is analyzed in connection with the energy dependence at fixed momentum transfer of the¹²C(e, e′) longitudinal and transverse responses. We compare our theoretical results with the available experimental data.     

On the triviality of λϕd4 theories and the approach to the critical point in d(−) > 4 dimensions

It is shown that one- and two-component λ|?|⁴ theories and non-linear σ-models in five or more dimensions approach free or generalized free fields in the continuum (scaling) limit, and that in four dimensions the same result holds, provided there is infinite field strength renormalization, as expected. Some critical exponents for the lattice theories in five or more dimensions are shown to be mean field. The main tools are Symanzik's polymer representation of scalar field theories and correlation inequalities.     

On the renormalized coupling constant and the susceptibility in φ44 field theory and the Ising model in four dimensions

We discuss the euclidean φ₄⁴ field theory, and the critical behavior in ferromagnetic systems in four dimensions. It is rigorously shown that there are at most logarithmic corrections to the mean field law in the behavior of the magnetic susceptibility _X = ΣS₂(0, x). Furthermore, if any such corrections are present in a continuum limit which is used to construct a φ₄⁴ field theory, the limiting theory would be non-interacting. Our analysis extends to ferromagnetic systems of variables which belong to the Griffiths-Simon class.     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.     

Relativistic Hartree-Bogoliubov description of the lithium isotopes

Here we report the development of the relativistic Hartree-Bogoliubov theory in coordinate space. Pairing correlations are taken into account by both density dependent force of zero range and finite range Gogny force. As a primary application the relativistic HB theory is used to describe the chain of Lithium isotopes reaching from ⁶Li to ¹¹Li. In contrast to earlier investigations within a relativistic mean field theory and a density dependent Hartree Fock theory, where the halo in ¹¹Li could only be reproduced by an artificial shift of the 1p _1/2 level close to the continuum limit, the halo is now reproduced in a self-consistent way without further modifications using the scattering of Cooper pairs to the 2s _1/2 level in the continuum. Excellent agreement with recent experimental data is observed.     

Horizontal interactions and their symmetries in GUTs

Using very general assumptions we find and discuss a large class of unified models with horizontal symmetries. We classify them and show on the basis of renormalisation group equations that the typical horizontal mass scale must be at least 10⁹–10¹³ GeV, depending on the model. A class of nonsupersymmetric theories with horizontal symmetries is discovered which predicts a proton lifetimeτ _p ≧10³³ and sin² θ _w ?0.23. It is also argued that supersymmetric unified models involving horizontal symmetries are unlikely to meet present experimental and theoretical requirements—contrary to ordinary supersymmetric theories without horizontal sector.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Cosmological difficulties for intermediate scales in superstring models

It is shown that primordial nucleosynthesis puts a severe limit M_I ≲ 10¹³ GeV on the intermediate mass scale in superstring theories. More stringent but also more uncertain limits can be obtained from the baryon-to-entropy ratio. All superstring models with more than three generations are ruled out.     

The :φ2: Field in theP(φ)2 model

Euclidean Field Theory techniques are used to study the Schwinger functions and characteristic function of the :φ²: field in evenP(φ)₂ models. The infinite volume limit is obtained for Half-Dirichlet boundary conditions by means of correlation inequalities. Analytic continuation yields Lorentz invariant Wightman functions. It is shown that, in the infinite volume limit, <:φ(x)²:>≧0 for both the Half and the Full-Dirichlet (λφ⁴)₂ model. This result also holds for a finite volume with periodic boundary conditions.     

Pregeometric quantum lattice: A general discussion

We put forward an idea that the fundamental, i.e. pregeometric, structure of spacetime is given by an abstract set, so called abstract simplicial complex ASC. Thus, at the pregeometric level there is no (smooth) spacetime manifold. However, we argue that the structure described by an abstract simplicial complex is dynamical. This dynamics is then assumed to ensure that ASC can be realized as a lattice on a four-dimensional manifold with the simplest topologies dominating.We rewrite the pregeometric model, which is quantized using euclidean path-integral formalism, in an exact way so that as a four-dimensional manifold with the simples topologies dominating. is done by definition. The first step in bringing the continuum into the arena is to build up a lattice on a four-dimensional manifold from a given ASC. In fact, we choose a specific lattice: The Regge calculus lattice, i.e. a piecewise linear (flat) metric spacetime manifold. Secondly, we introduce a smooth (C^∞) manifold (described by a metric tensor g_μν) to approximate the Regge calculus manifold (described by a metric tensor g_μν^RC).It turns out that after integrating (and summing) out all other degrees of freedom than the metric tensor field g_μν, the resulting continuum theory is nonlocal (as would be expected). However, it is our main point to show that the nonlocality is not very severe since it is only of finite range. We argue that the points in the introduced continuum which represent lattice points have so great quantum fluctuations that they are in a high temperature phase with no long-range correlations. In other words, although the effective action for the continuum formulation is not totally local, it is effectively so because it has only finite range nonlocalities. We can prove this kind of weak locality of the effective action by means of a general high-temperature theorem. Then we claim that the resulting local (or rather almost local) model with reparametrization invariance and g_μν as a field gives essentially the ordinary Einstein's gravity theory in the long wavelength limit.     

Effective field theory for massive gravitons and gravity in theory space

We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale M_Pl and a mass m_g, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Λ parametrically above m_g. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz-Pauli form of the Lagrangian, the presence or absence of the van Dam-Veltman-Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz-Pauli term are easily determined. The cutoff scales as Λ∼(m_g⁴M_Pl)^1/5 for the Fierz-Pauli theory, but can be raised to Λ∼(m_g²M_Pl)^1/3 in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions.     

On homogenization and scaling limit of some gradient perturbations of a massless free field

We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models. This article was processed by the author using the LAT_EX style filepljour1 from Springer-Verlag.     

Lattice (Φ 4)4 effective potential giving spontaneous symmetry breaking and the role of the Higgs mass

We present a critical reappraisal of the available results on the broken phase ofλ(Φ ⁴)₄ theory, as obtained from rigorous formal analyses and from lattice calculations. All the existing evidence is compatible with Spontaneous Symmetry Breaking but dictates a trivially free shifted field that becomes controlled by a quadratic hamiltonian in the continuum limit. As recently pointed out, this implies that the simple one-loop effective potential should become effectively exact. Moreover, the usual naive assumption that the Higgs mass-squaredm _h ² is proportional to its “renormalized” self-couplingλ _R is not valid outside perturbation theory: the appropriate continuum limit hasm _h finite and vanishingλ _R . A Monte Carlo lattice computation of theλ(Φ ⁴)₄ effective potential, both in the single-component and in theO(2)-symmetric cases, is shown to agree very well with the one-loop prediction. Moreover, its perturbative leading-log improvement (based on the concept ofλ _R ) fails to reproduce the Monte Carlo data. These results, while supporting in a new fashion the peculiar “triviality” of theλ(Φ ⁴)₄ theory, also imply that, outside perturbation theory, the magnitude of the Higgs mass does not give a measure of the observable interactions in the scalar sector of the standard model.     

Deformations of Fermionic Quantum Field Theories and Integrable Models

Considering the model of a scalar massive Fermion, it is shown that by means of deformation techniques it is possible to obtain all integrable quantum field theoretic models on two-dimensional Minkowski space which have factorizing S-matrices corresponding to two-particle scattering functions S ₂ satisfying S ₂(0) = ?1. Among these models there is for example the Sinh-Gordon model. Our analysis provides a complement to recent developments regarding deformations of quantum field theories. The deformed model is investigated also in higher dimensions. In particular, locality and covariance properties are analyzed.     

The appearance of critical dimensions in regulated string theories (II)

We supply numerical evidence for the existence of critical dimensions d_c1 and d_c2 gaussian model of a discretized string, between which the mean number of vertices in the world sheet diverges, and hence a continuum limit may exist. We also discuss the possibility of non-trivial continuum limits. In particular, by introducing an additional parameter into the model, we argue that non-trivial continuum limits can possibly be obtained only between two critical dimensions d′₀ and d₀. Furthermore, we give proofs of some lower bounds on determinants of combinatorial laplacians entering the models, which were announced in a previous paper.     

Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: Effects of branching ratio and homogeneous broadening

The Heitler-Ma damping theory is developed for a two level system in which the excited state is homogeneously, and irreversibly coupled to various continuum states with a total decay rate 1/τ. We give particular consideration to the channel consisting of a third, discrete, atomic level and a continuum of emitted photons, which simply corresponds to a spontaneous resonant Raman process. The theory applies to either a narrow, pulsed, laser beam, or injection of target atoms or molecules into a c.w. field.In this paper, we examine the t=∞ spectra as a function of field strength, detuning from resonance, and especially as a function of the upper state broadening, characterized by the branching ratio f_A=τ/t_A, where τ^-_A is the natural resonance fluorescence lifetime. For strong fields, we obtain the usual resonance fluorescence spectrum centered at the incident, pumping frequency, with two symmetric side bands displaced by the Rabi frequency. If f_A→0, the spectrum approaches the one-photon limit, with the height of the side bands equal to $\text{1}\text{2}$ that of the central peak and all of equal width. In this limit, the target predominantly decays into the Raman or other irreversible channels, and only a single laser photon contributes to the extremely weak resonance spectrum. At the opposite extreme, f_A→1, the target scatters many photons out of the laser-field before it is optically pumped into a non-interacting state and the spectrum exhibits the infinite cascade properties obtained by Mollow. The side bands become broadened, with a height equal to $\text{1}\text{3}$ of the central peak. In this theory, we obtain a more complete interpretation of the elastically scattered delta function, which is an artifact of the infinite lifetime of the atom in the usual two-level theories. In both limits of f_A, we obtain a Raman lineshape which is unchanged and is simply a function of the total width ?/τ of the excited state.