Published by Institute of Physics Publishing for SISSA/ISAS Received: May 14, 2004 Revised: August 24, 2004 Accepted: August 30, 2004 Is there a string theory landscape? JHEP08(2004)058 Tom Banks, Michael Dine and Elie Gorbatov Santa Cruz Institute for Particle Physics Santa Cruz CA 95064, U.S.A. Email: [email protected], [email protected], [email protected] Abstract: We examine recent claims of a large set of flux compacti¯cation solutions of string theory. We conclude that the arguments for AdS solutions are plausible. The analysis of meta-stable dS solutions inevitably leads to situations where long distance e®ective ¯eld theory breaks down. We then examine whether these solutions are likely to lead to a description of the real world. We conclude that one must invoke a version of the anthropic principle which insists on carbon based life forms. We explain why it is likely that this leads to a prediction of low energy supersymmetry breaking, but that many features of anthropically selected flux compacti¯cations are likely to disagree with experiment. Keywords: Superstrings and Heterotic Strings, Superstring Vacua. °c SISSA/ISAS 2004 http://jhep.sissa.it/archive/papers/jhep082004058 /jhep082004058.pdf Contents 1. Introduction 1 2. The proper use of e®ective lagrangians 3 2.1 The e®ective potential in string perturbation theory 5 3. Exploring the landscape of flux compacti¯cations 7 3.1 Detailed analysis of the discretuum 8 3.2 Vacuum tunneling and the existence of meta-stable dS states 10 4. Scienti¯c explanation in the discretuum 14 JHEP08(2004)058 4.1 An aside on continuous moduli spaces 15 4.2 Extremely light scalars? 16 5. The anthropic principle in flux compacti¯cations 17 5.1 The anthropic principle as a datum 17 5.2 The renormalization group and the anthropic principle 18 5.3 Bottom up approach to the phenomenology of the discretuum 21 5.4 Might the flux discretuum predict low energy supersymmetry? 24 5.5 Above the TeV scale: cosmological constraints 26 6. Conclusions 29 1. Introduction String Theory has always been plagued by a plethora of solutions which do not describe the real world. Early hopes that some of the highly supersymmetric (SUSic) moduli spaces of solutions would prove inconsistent once nonperturbative physics was understood, were dashed in a de¯nitive manner by the second superstring revolution. More recently there have been claims [1]{[7] that there is a large set of additional solutions of string theory, both SUSic and SUSY violating, with a discretuum of values of the cosmological constant. Susskind [8] has dubbed this discrete set of solutions the landscape of string theory, and suggested that anthropic arguments will have to be invoked to explain why the world we see is chosen from among this vast class of solutions. Douglas [9], has carried through what might be described as a prototype counting of such vacua. Ultimately, one might hope determine the distribution of couplings among the states of the landscape (what Douglas refers to as the \ensemble"). The purpose of this paper is to assess the arguments for the existence of the flux discretuum, as well as the prospects that any of these solutions could describe experimental physics. Our conclusions may be summarized as follows: the arguments for the existence of { 1 { AdS solutions, both SUSic and SUSY violating, appear plausible. The key to verifying their existence would be the discovery of dual CFT's for each of these solutions. Silverstein [10] has suggested a method for constructing these duals using brane/flux transitions. The meta-stable dS solutions are on much less ¯rm ground. Many, perhaps all, of them are likely to tunnel to Big Crunch cosmologies, where the low energy e®ective ¯eld theory technology, which is the primary tool for constructing these solutions, breaks down. We argue that these instabilities must be faced even if the amplitudes for them are much smaller than those for other decay modes. Furthermore, even in the absence of these instabilities, the putative stable state into which these solutions decay, is a Big Bang universe where e®ective ¯eld theory breaks down. Unless we develop techniques for understanding these singularities, there is no reliable evidence that these solutions are approximations to meaningful models of quantum gravity. The existence of a discretuum of stable or metastable states would force us to adopt JHEP08(2004)058 a new paradigm for scienti¯c explanation.1 We will discuss the possible paradigms, but, as we will explain, at least within the classes of flux vacua which have been considered up to now, the possibility that the real world is described by one of these states appears somewhat dim. One possible mode of explanation is to simply take a set of observed experimental facts, the gauge group, particle content and couplings of the Standard Model, the spectrum of fluctuations of the CMBR and the like, as given, and ask whether further predictions are possible. The second is to adopt an anthropic explanation. In the ¯rst case, as we will explain, it is essentially impossible to determine which is the \true vacuum" within the dis- cretuum. It is necessary to ask what (if any) subset of states is consistent with the observed experimental facts and whether the distribution of some as yet unmeasured quantities is highly peaked about some values (this viewpoint has been stated most forcefully by Dou- glas). The flux discretuum would appear to be a natural framework in which to implement the anthropic principle [12]. Indeed, it is clear that in the context of the landscape, this is the only way in which to explain the value of the cosmological constant, and probably several other quantities of low energy physics. But we will see that in the flux discretuum, such a viewpoint will almost inevitably lead to predictions which are false. We have previously presented [26] a variety of arguments against the utility of the anthropic principle in models of this type. In asking about anthropic vacuum selection, we distinguished two possibilities (what Dimopoulos has dubbed \habitability criteria"), which we will refer to as \Selection Principle A" and \Selection Principle B". Principle B selection principle asserts that any states which meet some very minimal criteria for the existence of life | the formation of structure in the universe, perhaps of stars, and the like, are acceptable. Selection principle A postulates the necessity for our own type of carbon based life. Principle B is a very reasonable constraint on mathematical theories of the universe, but is not very powerful. Principle A is potentially much more powerful, but requires one make assumptions which are very di±cult to justify. Given a collection of 1We are assuming, here, that there is not some sort of (possibly cosmological) selection principle which picks out one or a few states; in this case, there is not really a discretuum of states. An example of such a principle is the hypothesis of cosmological SUSY breaking [11]. { 2 { low energy gauge groups and representations, we simply do not understand enough about complex phenomena in (the various possible analogs of )low energy nuclear physics, much less (exo)biology, to assert that life is or is not possible. One of our points will be that even if we implement some form of principle A, the landscape is likely to make many incorrect predictions. In [26], we presented a variety of arguments against the utility of even principle A in models of the class exempli¯ed by the flux discretuum. In this paper we modify and extend those arguments, explaining why the predictions of principle A within the framework of these models , are likely to di®er widely from experimental values of many parameters. Apart from these negative comments, we note that there is one interesting prediction that the flux vacuum might make: low energy supersymmetry. In particular, we explain why the vast majority of vacua with cosmological constant and gauge hierarchy required by principle A, are likely to exhibit some form of low energy supersymmetry. This is not JHEP08(2004)058 necessarily a success; again, within the framework of principle A, our arguments suggest that the flux discretuum is not likely to agree with experiment. Some, though not all, of these observational problems are linked to the prediction of low energy SUSY. We begin our discussion in the next section by critically examining the notion of the low energy e®ective lagrangian, and the assumption that di®erent solutions of the same long wavelength e®ective ¯eld equations are related to the same model of quantum grav- ity. Section 3 is devoted to a discussion of AdS and (primarily) meta-stable dS solutions of long distance e®ective SUGRA, presenting our detailed argument for the conclusions adumbrated above. In section 4 we turn to phenomenological questions and the anthropic principle. Section 5 reiterates our conclusions. The system behind our exposition is to present a series of potential problems with the notion of a string theory landscape. These problems are ordered from fundamental to phenomenological. At each stage of the expo- sition, we ignore the objections of the previous stage, but the reader is certainly expected to keep them in mind. 2. The proper use of e®ective lagrangians To explain our point of view about e®ective lagrangians in string theory, we will have to introduce some non-standard terminology. We will use the term long wavelength e®ective lagrangian to describe the object that is conventionally derived from quantum ¯eld theory or string S-matrices. In [15], one of the authors (TB) argued that, when supplemented with asymptotic boundary conditions in space-time, the long wavelength e®ective lagrangian in a model of quantum gravity also carries information about the high energy spectrum of the theory, via the properties of its black hole solutions.