Peter E. Haagensen

SCHEME INDEPENDENCE AND THE EXACT RENORMALIZATION GROUP

Abstract We compute critical exponents in a Z2 symmetric scalar field theory in three dimensions, using Wilsons exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved.

Spatial geometry of the electric field representation of non-abelian gauge theories☆

A unitary transformation �[E] = exp(i[E]/g)F[E] is used to simplify the Gauss law constraint of non-abelian gauge theories in the electric field representation. This leads to an unexpected geometrization because ! a � �[E]/�E ai transforms as a (composite) connection. The geometric information in ! a is transferred to a gauge invariant spatial connection i and torsion by a suitable choice of basis vectors for the adjoint representation which are constructed from the electric field E ai . A metric is also constructed from E ai . For gauge group SU(2), the spatial geometry is the standard Riemannian geometry of a 3-manifold, and for SU(3) it is a metric preserving geometry with both conventional and unconventional torsion. The transformed Hamiltonian is local. For a broad class of physical states, it can be expressed entirely in terms of spatial geometric, gauge invariant variables. The canonical commutation relations and Gauss law constraint of Hamiltonian gauge theories in temporal gauge are invariant under spatial diffeomorphisms of the canonical variables A a (x) and E ai (x). This local GL(3) symmetry is broken in the Hamiltonian in a simple way because of the appearance of the Cartesian metric δij of flat space, and the energy density transforms as a GL(3) tensor density. In this paper we discuss a formulation of non-abelian gauge theories in which the Gauss law constraint is easily implemented and the Hamiltonian is expressed in terms of variables which are gauge invariant or covariant and also geometric, i.e. they are GL(3) tensors, connections or curvatures. The resulting theory has an elegant mathematical structure but it is far from clear that the spatial geometry will be helpful for dynamical calculations or offer any advantages over such well-developed approaches as lattice gauge

Gradient Flows From An Approximation To The Exact Renormalization Group

Through appropriate projections of an exact renormalization group equation, we study fixed points, critical exponents and nontrivial renormalization group flows in scalar field theories in 2 < D < 4. THE STANDARD UPPER CRITICAL DIMENSIONS D k = 2k/(k − 1), k = 2, 3, 4,... appear naturally encoded in our formalism, and for dimensions smaller but very close to d k our results match the e-expansion. Within the coupling constant subspace of mass and quartic couplings and for any d, we find a gradient flow with two fixed points determined by a positive-definite metric and a c-function which is monotonically decreasing along the flow

Differential renormalization of massive quantum field theories

Abstract We extend the method of differential renormalization to massive quantum field theories, treating in particular λϕ 4 -theory and QED. As in the massless case, the method proves to be simple and powerful, and we are able to find, in particular, compact explicit coordinate space expressions for the finite parts of two notably complicated diagrams, namely, the two-loop two-point function in λϕ 4 and the one-loop vertex in QED.

A Comprehensive Coordinate Space Renormalization of Quantum Electrodynamics to Two-Loop Order

We develop a coordinate space renormalization of massless quantum electrodynamics using the powerful method of differential renormalization. Bare one-loop amplitudes are finite at non-coincident external points, but do not accept a Fourier transform into momentum space. The method provides a systematic procedure to obtain one-loop renormalized amplitudes with finite Fourier transforms in strictly four dimensions without the appearance of integrals or the use of a regulator. Higher loops are solved similarly by renormalizing from the inner singularities outwards to the global one. We compute all one- and two-loop 1PI diagrams, run renormalization group equations on them. and check Ward identities. The method furthermore allows us to discern a particular pattern of renormalization under which certain amplitudes are seen not to contain higher-loop leading logarithms. We finally present the computation of the chiral triangle showing that differential renormalization emerges as a natural scheme to tackle [gamma][sub 5] problems.

Duality transformations away from conformal points

Abstract Target space duality transformations are considered for bosonic sigma models and strings away from RG fixed points. A set of consistency conditions are derived, and are seen to be nontrivially satisfied at one-loop order for arbitrary running metric, antisymmetric tensor and dilaton backgrounds. Such conditions are sufficiently stringent to enable an independent determination of the sigma model beta functions at this order.

Differential renormalization of the Wess-Zumino model

We apply the recently developed method of differential renormalization to the Wess-Zumino model. From the explicit calculation of a finite, renormalized effective action, the β-function is computed to three loops and is found to agree with previous existing results. As a further, non-trivial check of the method, the Callan-Symanzik equations are also verified to that loop order. Finally, we argue that differential renormalization presents advantages over other superspace renormalization methods, in that it avoids both the ambiguities inherent to supersymmetric regularization by dimensional reduction (SRDR), and the complications of virtually all other supersymmetric regulators.

DIFFERENTIAL RENORMALIZATION FOR CURVED SPACE AND FINITE TEMPERATURE

We derive, based only on simple principles of renormalization in coordinate space, closed renormalized amplitudes and renormalization group constants at one- and two-loop orders for scalar field theories in general backgrounds. This is achieved through a renormalization procedure we develop exploiting the central idea behind differential renormalization, which needs as the only inputs the propagator and the appropriate Laplacian for the backgrounds in question. We work out this coordinate space renormalization in some detail, and subsequently back it up with specific calculations for scalar theories both on curved backgrounds, manifestly preserving diffeomorphism invariance, and at finite temperature.

THE SUPERSYMMETRIC CURCI-PAFFUTI RELATION AND VANISHING OF THE DILATON BETA FUNCTION IN THE N=2 SUSY SIGMA MODEL

We extend the Curci-Paffuti relation of bosonic sigma models to the supersymmetric case. In the N=1 model, a similar relation is found, while in the N=2 model, a vanishing result ensues for the dilaton β-function. One contribution to the dilaton β-function in the N=2 model is identified as a previous result of Grisaru and Zanon; however, if we remain within a minimal subtraction scheme, other terms coming from finite subtractions appear which precisely cancel that and give a vanishing result. This is in agreement with a recent result of Jack and Jones.

THE LORENTZ CHERN-SIMONS FORM IN HETEROTIC GREEN-SCHWARTZ SIGMA MODELS

By studying the 1-loop effective action of the heterotic Green-Schwartz sigma model, we identify the only source of anomaly in the target local Lorentz symmetry, and verify that it is canceled by the inclusion of a Lorentz Chern-Simons three-form in the definition of the supergravity field HABC.