Lambda theories of effective lambda models

A longstanding open problem is whether there exists a non-syntactical model of untyped lambda-calculus whose theory is exactly the least equational lambda-theory (=Lb). In this paper we make use of the Visser topology for investigating the more general question of whether the equational (resp. order) theory of a non syntactical model M, say Eq(M) (resp. Ord(M)) can be recursively enumerable (= r.e. below). We conjecture that no such model exists and prove the conjecture for several large classes of models. In particular we introduce a notion of effective lambda-model and show that for all effective models M, Eq(M) is different from Lb, and Ord(M) is not r.e. If moreover M belongs to the stable or strongly stable semantics, then Eq(M) is not r.e. Concerning Scott's continuous semantics we explore the class of (all) graph models, show that it satisfies Lowenheim Skolem theorem, that there exists a minimum order/equational graph theory, and that both are the order/equ theories of an effective graph model. We deduce that no graph model can have an r.e. order theory, and also show that for some large subclasses, the same is true for Eq(M).Comment: 15 pages, accepted CSL'0

Effective lambda-models vs recursively enumerable lambda-theories

A longstanding open problem is whether there exists a non syntactical model of the untyped lambda-calculus whose theory is exactly the least lambda-theory (l-beta). In this paper we investigate the more general question of whether the equational/order theory of a model of the (untyped) lambda-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of lambda-calculus calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be l-beta or l-beta-eta. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is minimum among all theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34

Unconventional low-energy SUSY from warped geometry

Supersymmetric models with a warped fifth spatial dimension can solve the hierarchy problem, avoiding some shortcomings of non-supersymmetric constructions, and predict a plethora of new phenomena at typical scales Lambda not far from the electroweak scale (Lambda ~ a few TeV). In this paper we derive the low-energy effective theories of these models, valid at energies below Lambda. We find that, in general, such effective theories can deviate significantly from the Minimal Supersymmetric Standard Model (MSSM) or other popular extensions of it, like the NMSSM: they have non-minimal Kaehler potentials (even in the Mp -> \infty limit), and the radion is coupled to the visible fields, both in the superpotential and the Kaehler potential, in a non-trivial (and quite model-independent) fashion. The corresponding phenomenology is pretty unconventional, in particular the electroweak breaking occurs in a non-radiative way, with tan beta \simeq 1 as a quite robust prediction, while the mass of the lightest Higgs boson can be as high as ~ 700 GeV.Comment: 53 pages, 2 ps figure

Renormalization Group Invariance of Exact Results in Supersymmetric Gauge Theories

We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff, provided appropriate changes are made to the bare coupling constants in the Lagrangian. We first pose a puzzle on how a quantum modified constraint (such as Pf(Q^i Q^j) = \Lambda^{2(N+1)} in SP(N) theories with N+1 flavors) can be RG invariant, since the bare fields Q^i receive wave function renormalization when one changes the ultraviolet cutoff, while we naively regard the scale \Lambda as RG invariant. The resolution is that \Lambda is not RG invariant if one sticks to canonical normalization for the bare fields as is conventionally done in field theory. We derive a formula for how \Lambda must be changed when one changes the ultraviolet cutoff. We then compare our formula to known exact results and show that their consistency requires the change in \Lambda we have found. Finally, we apply our result to models of supersymmetry breaking due to quantum modified constraints. The RG invariance helps us to determine the effective potential along the classical flat directions found in these theories. In particular, the inverted hierarchy mechanism does not occur in the original version of these models.Comment: LaTeX, 26 page

Phenomenology of dark energy: exploring the space of theories with future redshift surveys

We use the effective field theory of dark energy to explore the space of modified gravity models which are capable of driving the present cosmic acceleration. We identify five universal functions of cosmic time that are enough to describe a wide range of theories containing a single scalar degree of freedom in addition to the metric. The first function (the effective equation of state) uniquely controls the expansion history of the universe. The remaining four functions appear in the linear cosmological perturbation equations, but only three of them regulate the growth history of large scale structures. We propose a specific parameterization of such functions in terms of characteristic coefficients that serve as coordinates in the space of modified gravity theories and can be effectively constrained by the next generation of cosmological experiments. We address in full generality the problem of the soundness of the theory against ghost-like and gradient instabilities and show how the space of non-pathological models shrinks when a more negative equation of state parameter is considered. This analysis allows us to locate a large class of stable theories that violate the null energy condition (i.e. super-acceleration models) and to recover, as particular subsets, various models considered so far. Finally, under the assumption that the true underlying cosmological model is the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) scenario, and relying on the figure of merit of EUCLID-like observations, we demonstrate that the theoretical requirement of stability significantly narrows the empirical likelihood, increasing the discriminatory power of data. We also find that the vast majority of these non-pathological theories generating the same expansion history as the $\Lambda$CDM model predict a different, lower, growth rate of cosmic structures.Comment: v1: 28 pages, 20 pdf figures. v2: 29 pages, minor improvements in the text, figures improve

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 Mpl and a mass mg, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Lambda parametrically above mg. 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 Lambda ~ (mg^4 Mpl)^(1/5) for the Fierz-Pauli theory, but can be raised to Lambda ~ (mg^2 Mpl)^(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.Comment: 22 pages, 7 diagrams; references and some clarifying comments added, typos correcte