On the Fixed-Point Structure of Scalar Fields

In a recent Letter (K.Halpern and K.Huang, Phys. Rev. Lett. 74 (1995) 3526), certain properties of the Local Potential Approximation (LPA) to the Wilson renormalization group were uncovered, which led the authors to conclude that $D>2$ dimensional scalar field theories endowed with {\sl non-polynomial} interactions allow for a continuum of renormalization group fixed points, and that around the Gaussian fixed point, asymptotically free interactions exist. If true, this could herald very important new physics, particularly for the Higgs sector of the Standard Model. Continuing work in support of these ideas, has motivated us to point out that we previously studied the same properties and showed that they lead to very different conclusions. Indeed, in as much as the statements in hep-th/9406199 are correct, they point to some deep and beautiful facts about the LPA and its generalisations, but however no new physics.Comment: Typos corrected. A Comment - to be published in Phys. Rev. Lett. 1 page, 1 eps figure, uses LaTeX, RevTex and eps

Manifestations of the Galactic Center Magnetic Field

Several independent lines of evidence reveal that a relatively strong and highly ordered magnetic field is present throughout the Galaxy's central molecular zone (CMZ). The field within dense clouds of the central molecular zone is predominantly parallel to the Galactic plane, probably as a result of the strong tidal shear in that region. A second magnetic field system is present outside of clouds, manifested primarily by a population of vertical, synchrotron-emitting filamentary features aligned with the field. Whether or not the strong vertical field is uniform throughout the CMZ remains undetermined, but is a key central issue for the overall energetics and the impact of the field on the Galactic center arena. The interactions between the two field systems are considered, as they are likely to drive some of the activity within the CMZ. As a proxy for other gas-rich galaxies in the local group and beyond, the Galactic center region reveals that magnetic fields are likely to be an important diagnostic, if not also a collimator, of the flow of winds and energetic particles out of the nucleus.Comment: To appear in "LESSONS FROM THE LOCAL GROUP" - A Conference in Honour of David Block and Bruce Elmegreen, eds: Freeman, K.C., Elmegreen, B.G., Block, D.L. & Woolway, M. (SPRINGER: NEW YORK

Elements of the Continuous Renormalization Group

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons.Comment: Invited lectures at the Yukawa International Seminar '97. 20 pages including 6 eps figs. LaTeX. PTPTeX style files include

Renormalization group properties of the conformal sector: towards perturbatively renormalizable quantum gravity

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in $\hbar$ and evanescent. For $\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale $L$, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with spacetimes of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton's constant but non-perturbative in $\hbar$.Comment: 52 pages, 4 figures; fixed typos; improved explanation of the sign of V, and the use of Sturm-Liouville theory. To be publ in JHE

The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all $\eta\ge.02$, finding the expected first ten FPs, and {\sl only} these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2\% and 33\%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory.Comment: Note added on "shadow operators". Version to be published in Phys. Lett.