Noncommutative induced gauge theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ⁴-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.     

Renormalisation of <Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript>-Theory on Noncommutative ℝ<Superscript>4</Superscript> in the Matrix Base

We prove that the real four-dimensional Euclidean noncommutative ⁴-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative ⁴. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.     

Three loop gauge β-function for the most general single gauge-coupling theory

We calculate the three loop contribution to the β-function of the gauge coupling constant in a general, anomaly-free, renormalisable gauge field theory involving a single gauge coupling using the background field method in the scheme.     

Monte Carlo study of a lattice gauge theory with a radially varied Higgs field

The phase structure of a gauge-scalar (Higgs) field system is studied by Monte Carlo simulations without freezing the radial mode of the scalar field. We consider Z₂ lattice gauge theory coupled to a Higgs field which is approximated by a discrete real one. Most of our analysis is done on a 4⁴ lattice. We find that the phase diagram of our model consists of three distinct phases, Higgs and confined regions being divided by a phase boundary. This phase structure forms a contrast with that presented in the model with a fixed-length Higgs field.     

On the multi trace superpotential and corresponding matrix model

We study supersymmetric U(N) gauge theory coupled to an adjoint scalar superfield with a cubic superpotential containing a multi trace term. We show that the field theory results can be reproduced from a matrix model whose potential is given in terms of a linearized potential obtained from the gauge theory superpotential by adding some auxiliary non-dynamical field. Once we get the effective action from this matrix model we could integrate out the auxiliary field getting the correct field theory results.Received: 2 May 2003, Published online: 18 December 2003     

Triviality pursuit: Can elementary scalar particles exist?

Great effort is presently being expended in the search for elementary scalar “Higgs” particles. These particles have yet to be observed. The primary justification for this search is the theoretically elegant Higgs-Kibble mechanism, in which the interactions of elemetary scalars are used to generate gauge boson masses in a quantum field theory. However, strong evidence suggests that at least a pure φ⁴ scalar field theory is trivial or noninteracting. Should this triviality persist in more complicated systems such as the standard model of the weak interaction, the motivation for looking for Higgs particles would be seriously undermined. Alternatively, the presence of gauge and fermion fields can rescue a pure scalar theory from triviality. Phenomenological constraints (such as a bounded or even predictable Higgs mass) may then be implied. In this report the evidence for triviality in various field theories is reviewed, and the implications for high energy physics are discussed.     

Generalized gauge fields and applications to weak interactions

Yang-Mills' field is generalized to possess a nontrivial scalar part. The most general transformations for such a field under the 3-parameter isotopic gauge transformation is obtained. Using this generalized gauge field, a gauge invariant Lagrangian is constructed within the framework of the quark model. Interactions for spin-1 as well as for spin-0 are generated. As a further application a weak interaction theory mediated by the generalized gauge (boson) field is formulated. The entire weak interactions are generated in two halfs; the hadron-boson interaction is generated according to Yang-Mills' trick using the generalized gauge field and the other half (boson-lepton, etc.) is then generated by making use of the scalar part of the gauge fields according to the conventional pion gauge principle. The effective Lagrangian is then found to be mediated by the effective propagators which fall off as p⁻² at high momenta; the unitarity of the theory can thereby be insured. Universality in weaker sense than the usual one is applied to the intermediate bosons; our theory for β-decay then reduces to Cabibbo's at low energy.     

Higher derivatives in quantum cosmology: (I). The isotropic case

This paper is the first in a series that investigates the effects on cosmology of curvature-squared terms which are bound to appear in the effective action whether or not gravity is perturbatively finite, is cut off by nonperturbative effects, or is made renormalisable by the addition of curvature squared terms to the bare action. We examine how these terms affect the recent proposal that the quantum state of the universe is defined by a path integral over compact metrics. In this paper we consider a simple minisuperspace model of an isotropic universe. In such a model the C-squared term in the action plays no role while the R-squared term behaves like a massive scalar field. The wave function of the universe can be interpreted as corresponding in the classical limit to a family of solutions that start out with a long period of exponential expansion and then go over to a matter dominated era.     

Gaussian effective potential for the U(1) Higgs model

In order to investigate the Higgs mechanism nonperturbatively, we compute the Gaussian effective potential of the U(1) Higgs model (“scalar electrodynamics”). We show that the same simple result is obtained in three different formalisms. A general covariant gauge is used, with Landau gauge proving to be optimal. The renormalization generalizes the “autonomous” renormalization for λ?⁴ theory and requires a particular relationship between the bare gauge coupling e _B and the bare scalar self-coupling λ _B. When both couplings are small, then λ is proportional to e⁴ and the scalar/vector mass-squared ratio is of order e², as in the classic 1-loop analysis of Coleman and Weinberg. However, as λ increases, e reaches a maximum value and then decreases, and in this “nonperturbative” regime the Higgs scalar can be much heavier than the vector boson. We compare our results to the autonomously renormalized 1-loop effective potential, finding close agreement in the physical predictions. The main phenomenological implication is a Higgs mass of about 2 TeV.     

Upper bound estimate for the Higgs mass from the lattice regularized Weinberg-Salam model

There are indications that, as a consequence of the trivality of the scalar φ⁴ theory, the Weinberg-Salam model is massive free field theory. However, considering theWS model as an effective field theory, an upper bound for the Higgs mass can exist. We have studied this phenomenon by simulating the SU(2) gauge Higgs model on a lattice. The results indicate that R_max = m_H/m_W|_max = 9.3 ± 1. However, serious finite size effects make it unfeasible to explore the region m_H ⩽ Λ_cutoff with Monte Carlo simulation.     

Vacuum instability in scalar field theories

The class of scalar field theories with interaction gφ^2N?1, are studied using the semi-classical approximation. The imaginary part of the vertex functions generated by tunnelling out of the metastable ground state is calculated to first order. Using this result, the leading asymptotic behaviour of the renormalisation group β function for φ³ field theory is obtained in six dimensions. The validity of this result is discussed in view of the extra singularities which appear when the theory is just renormalisable. The structure of the perturbation expansion for n component φ³ theory is also discussed, and cases in which these theories yield perturbation expansions which are Borel summable, are pointed out.     

Dual models for non-hadrons

The possibility of describing particles other than hadrons (leptons, photons, gauge bosons, gravitons, etc.) by a dual model is explored. The Virasoro-Shapiro model is studied first, interpreting the massless spin-two state of the model as a graviton. We prove that in the limit of zero slope (with g_vs²α′ held fixed) one obtains the Einstein theory of gravitation accompanied by a massless scalar field. Next, the Veneziano model is studied for small slope as an expansion in powers of α′. It is known from previous work that the zeroth order term is precisely the Yang-Mills theory of a multiplet of massless vector bosons. We show that there are order α′ terms arising both from the dual tree and loop graphs. The former constitutes a relatively unimportant modification of the Yang-Mills theory, whereas the latter involves the coupling of the massless scalar and graviton states of the Virasoro-Shapiro model. Thus one may take the point of view that gravity arises as a unitarization effect in a dual unified theory of electromagnetism and weak interactions. In order to obtain the correct values for the electric charge and Newton's constant it is necessary that α′ ? 10^?34 GeV^?2 The coupling of massless scalar states is also studied.     

Multidimensional Unified Theories

An extended spacetime, M^4+N, is a Riemannian (4 + N)-dimensional manifold which admits an N-parameter group G of (spacelike) isometries and is such that ordinary spacetime M⁴ is the space M^4+N/G of the equivalence classes under G-transformations of M^4+N. A multidimensional unified theory (MUT) is a dynamical theory of the metric tensor on M^4+N, the metric being determined from the Einstein-Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M^4+N. A MUT is an extension of the Cho-Freund generalization of Jordan's five-dimensional theory. A MUT can be faithfully translated in four-dimensional language: as a theory on M⁴, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on M^4+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt-Trautman generalization of Kaluza's five-dimensional theory; in four-dimensional language, this is the theory of Yang-Mills gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang-Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly-constrained MUT leads to bag-type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.     

General relativity as a conformally invariant scalar gauge field theory

The global symmetry implied by the fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes Conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form.     

Temperature dependence of coupling constants

We study the temperature-dependence of coupling constants at the one-loop level for massive ?⁴ theory and massive scalar electrodynamics (SED). It is found that the scalar coupling constant λ for m² > 0 decreases with temperature leading to a phase transition to a non-interacting phase. In a model with m² < 0, λ increases as 1n T. The gauge coupling constant of SED increases uniformly with temperature.     

Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

On quasi-self-dual fields in N=4 supersymmetric Yang-Mills theory

A general representation for a quasi-self-dual gauge field which is a generalization of the 't Hooft ansatz is obtained in the Euclidean conformally invariant Yang-Mills theory with a sextuplet of real scalar fields-the boson sector of N=4 supersymmetric Yang-Mills theory. The existence of an O(4)-symmetric quasi-self-dual solution in the region g ²>0 is proved. An explicit example of the quasi-self-dual instanton in the region g ²<0 is constructed.     

Supersymmetry and classical solutions

We obtain classical solutions to the field equations of the massless supersymmetric Wess-Zumino model and to the field equations of the interacting SU(2) gauge supermultiplet. This is done by applying finite supersymmetry transformations to the known solutions of the scalar field equation with ?⁴ interaction and the Yang-Mills field equations. The relevance of supersymmetry to the solution of classical field equations involving anticommuting fermion fields is discussed.     

QCD calculations in the light-cone gauge

The non-singlet quark structure function is calculated in the leading logarithm approximation in an axial gauge with n² = 0, the light-cone gauge. The choice n² = 0 leads to a simple identity for loop integrals involving the extra n · k denominators. We compare the results graph by graph with both Feynman gauge QCD and a scalar gluon theory. The leading diagrams are the same “rainbow” diagrams as for the case of the scalar theory.The techniques are also applied to quark-quark scattering at large transverse momentum. The leading diagrams have the same dressed ladder-form factor structure.