Entry and Return times distribution

This is a review article on the distributions of entry and return times in dynamical systems which discusses recent results for systems of positive entropy.Comment: To appear in "Dynamical Systems: An International Journal dedicated to the Statistical Properties of Dynamical Systems

Renormalization Group and Effective Potential in Classically Conformal Theories

We derive a general formula for the RG improved effective (Coleman-Weinberg) potential for classically conformal models, applying it to several examples of physical interest, and in particular a model of QCD coupled via quarks to a colorless scalar field. The closed form expressions allow us to discuss the range of validity of the effective potential as well as the issue of `large logarithms' in a way different from previous such analyses. Remarkably, in all examples considered, convexity of the effective potential is restored by the RG improvement, or otherwise the potential becomes unstable. In the former case, symmetry breaking becomes unavoidable due to the appearance of an infrared barrier $\Lambda_{IR}$, which hints at a so far unsuspected link between $\Lambda_{QCD}$ and the scale of electroweak symmetry breaking.Comment: 17 pages, 2 figures, formulas corrected, references adde

Conformal Anomaly and Off-Shell Extensions of Gravity

The gauge dependence of the conformal anomaly for spin 3/2 and spin 2 fields in non-conformal supergravities has been a long standing puzzle. In this Letter we argue that the `correct' gauge choice is the one that follows from requiring all terms that would imply a violation of the Wess-Zumino consistency condition to be absent in the counterterm, because otherwise the usual link between the anomaly and the one-loop divergence becomes invalid. Remarkably, the `good' choice of gauge is the one that confirms our previous result that a complete cancellation of conformal anomalies in D=4 can only be achieved for N-extended (Poincar\'e) supergravities with $N\geq 5$

Conformal invariance from non-conformal gravity

We discuss the conditions under which classically conformally invariant models in four dimensions can arise out of non-conformal (Einstein) gravity. As an `existence proof' that this is indeed possible we show how to derive N=4 super Yang Mills theory with any compact gauge group G from non-conformal gauged N=4 supergravity as a special flat space limit. We stress the role that the anticipated UV finiteness of the (so far unknown) underlying theory of quantum gravity would have to play in such a scheme, as well as the fact that the masses of elementary particles would have to arise via quantum gravitational effects which mimic the conformal anomalies of standard (flat space) UV divergent quantum field theory.Comment: 18 page

Conformal Anomalies and Gravitational Waves

We argue that the presence of conformal anomalies in gravitational theories can lead to observable modifications to Einstein's equations via the induced anomalous effective actions, whose non-localities can overwhelm the smallness of the Planck scale. The fact that no such effects have been seen in recent cosmological or gravitational wave observations therefore imposes strong restrictions on the field content of possible extensions of Einstein's theory: all viable theories should have vanishing conformal anomalies. We then show that a complete cancellation of conformal anomalies in D=4 for both the $C^2$ invariant and the Euler (Gauss-Bonnet) invariant can only be achieved for N-extended supergravities with $N \geq 5$, as well as for M theory compactified to four dimensions.Comment: 4 page

Conformal Symmetry and the Standard Model

We re-examine the question of radiative symmetry breaking in the standard model in the presence of right-chiral neutrinos and a minimally enlarged scalar sector. We demonstrate that, with these extra ingredients, the hypothesis of classically unbroken conformal symmetry, besides naturally introducing and stabilizing a hierarchy, is compatible with all available data; in particular, there exists a set of parameters for which the model may remain viable even up to the Planck scale. The decay modes of the extra scalar field provide a unique signature of this model which can be tested at LHC.Comment: 13 pages, 6 figures, version to be published in Phys. Lett.

Modular realizations of hyperbolic Weyl groups

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group