Generalized Gaugino Condensation in Super Yang-Mills Theories: Discrete R-Symmetries and Vacua

One can define generalized models of gaugino condensation as theories which dynamically break a discrete R-symmetry, but do not break supersymmetry. We consider general examples consisting of gauge and matter fields, and the minimal number of gauge singlet fields to avoid flat directions in the potential. We explore which R-symmetries can arise, and their spontaneous breaking. In general, we find that the discrete symmetry is $\mathbb{Z}_{2b_0R}$ and the number of supersymmetric vacua is $b_0$, where $b_0$ is the coefficient of the one-loop beta function. Results are presented for various groups, including $SU(N_c), SO(N_c), Sp(2N_c)$, and $G_2$, for various numbers of flavors, $N_f$, by several methods. This analysis can also apply to the other exceptional groups, and thus all simple Lie groups. We also comment on model building applications where a discrete R-symmetry, broken by the singlet vevs, can account for $\mu$-type terms and allow a realistic Higgs spectrum naturally.Comment: 16 page

Chaotic Inflation from Nonlinear Sigma Models in Supergravity

We present a common solution to the puzzles of the light Higgs or quark masses and the need for a shift symmetry and large field values in high scale chaotic inflation. One way to protect, for example, the Higgs from a large supersymmetric mass term is if it is the Nambu-Goldstone boson (NGB) of a nonlinear sigma model. However, it is well known that nonlinear sigma models (NLSMs) with nontrivial K\"ahler transformations are problematic to couple to supergravity. An additional field is necessary to make the K\"ahler potential of the NLSM invariant in supergravity. This field must have a shift symmetry --- making it a candidate for the inflaton (or axion). We give an explicit example of such a model for the coset space $SU(3)/SU(2) \times U(1)$, with the Higgs as the NGB, including breaking the inflaton's shift symmetry and producing a chaotic inflation potential. This construction can also be applied to other models, such as one based on $E_7/SO(10) \times U(1) \times U(1)$ which incorporates the first two generations of (light) quarks as the Nambu-Goldstone multiplets, and has an axion in addition to the inflaton. Along the way we clarify and connect previous work on understanding NLSMs in supergravity and the origin of the extra field (which is the inflaton here), including a connection to Witten-Bagger quantization. This framework has wide applications to model building; a light particle from a NLSM requires, in supergravity, exactly the structure for chaotic inflaton or an axion.Comment: 5 page

Discrete R Symmetries and Low Energy Supersymmetry

If nature exhibits low energy supersymmetry, discrete (non-$Z_2$) R symmetries may well play an important role. In this paper, we explore such symmetries. We generalize gaugino condensation, constructing large classes of models which are classically scale invariant, and which spontaneously break discrete R symmetries (but not supersymmetry). The order parameters for the breaking include chiral singlets. These simplify construction of models with metastable dynamical supersymmetry breaking. We explain that in gauge mediation, the problem of the cosmological constant makes "retrofitting" particularly natural -- almost imperative. We describe new classes of models, with interesting scales for supersymmetry breaking, and which allow simple solutions of the $\mu$ problem. We argue that models exhibiting such R symmetries can readily solve not only the problem of dimension four operators and proton decay, but also dimension five operators. On the other hand, in theories of "gravity mediation", the breaking of R symmetry is typically of order $M_p$, R parity is required to suppress dimension four $B$ and $L$ violating operators, and dimension five operators remain problematic.Comment: 17 pages, latex; references to earlier work added; typos fixed, minor corrections and revisions, journal versio

Axions in the Landscape and String Theory

While axions seem ubiquitous in critical string theories, whether they might survive in any string theoretic description of nature is a difficult question. With some mild assumptions, one can frame the issues in the case that there is an approximate supersymmetry below the underlying string scale. The problem of axions is then closely tied to the question of how moduli are fixed. We consider, from this viewpoint, the possibility that supersymmetry is broken at an intermediate scale, as in "gravity mediation," at a low scale, as in gauge mediation, and at a very high scale, to model the possibility that there is no low energy supersymmetry. Putative mechanisms for moduli fixing can then be systematically classified, and at least for intermediate and high scale breaking, light axions appear plausible. In the course of this work, we are lead to consider aspects of moduli fixing and supersymmetry breaking, and we revisit the possibility of very large extra dimensions.Comment: 18 pages; v2, updated and added reference