We develop techniques to classify D- and F-flat directions for N=1 supersymmetric string vacua of the perturbative heterotic string theory, which possess an anomalous U(1) gauge group at the tree level. Genus-one corrections generate a Fayet-Iliopoulos term for the D-term of U(1)_A, which is canceled by non-zero vacuum expectation values (VEVs) of certain massless multiplets in such a way that the anomalous U(1) is broken, while maintaining the D- and F-flatness of the effective field theory. A systematic analysis of flat directions is given for non-zero VEVs of non-Abelian singlets, and the techniques are illustrated for a specific model. The approach sets the stage to classify the D- and F-flat directions for a large class of perturbative string vacua. This classification is a prerequisite to address systematically the phenomenological consequences of these models.Comment: 26 page

We address intermediate scales within a class of string models. The intermediate scales occur due to the SM singlets $S_i$ acquiring non-zero VEVs due to radiative breaking; the mass squared $m_i^2$ of $S_i$ is driven negative at ${\mu }_{\mathrm{rad}}$ due to O(1) Yukawa couplings of $S_i$ to exotic particles (calculable in a class of string models). The actual VEV of $S_i$ depends on the relative magnitude of the non-renormalizable terms of the type $S_i^{K+3\mathrm{}}{/M}^K$ in the superpotential. We mainly consider the case in which the $S_i$ are charged under an additional non-anomalous ${U\left(1\right)\mathrm{}}^{\prime }$ gauge symmetry and the VEVs occur along $F$- and $D$-flat directions. We explore various scenarios in detail, depending on the type of Yukawa couplings to the exotic particles and on the initial boundary values of the soft SUSY breaking parameters. We then address the implications of these scenarios for the μ parameter and the fermionic masses of the standard model.

From the “top-down” approach we investigate physics implications of the class of D- and F-flat directions formed from non-Abelian singlets which are proved flat to all orders in the nonrenormalizable superpotential, for a prototype quasi-realistic free fermionic string model with the standard model gauge group and three families (CHL5). These flat directions have at least an additional ${U\left(1\right)\mathrm{}}^{\prime }$ unbroken at the string scale. For each flat direction, the complete set of effective mass terms and effective trilinear superpotential terms in the observable sector are computed to all orders in the VEV’s of the fields in the flat direction. The “string selection rules” disallow a large number of couplings allowed by gauge invariance, resulting in a massless spectrum with a large number of exotics, in most cases excluded by experiment, thus signifying a generic flaw of these models. Nevertheless, the resulting trilinear couplings of the massless spectrum possess a number of interesting features which we analyze for two representative flat directions: for the fermion texture, baryon- and lepton-number violating couplings, R-parity breaking, non-canonical $\mu$ terms, and the possibility of electroweak and intermediate scale symmetry breaking scenarios for ${U\left(1\right)\mathrm{}}^{\prime }.$ The gauge coupling predictions are obtained in the electroweak scale case. Fermion masses possess $t-b$ and $\tau -\mu$ universality, with the string scale Yukawa couplings g and $g/\sqrt{2},$ respectively. Fermion textures are present for certain flat directions, but only in the down-quark sector. Baryon- and lepton-number violating couplings can trigger proton decay, $N-\overline{N}$ oscillations, leptoquark interactions and R-parity violation, leading to the absence of a stable LSP.

Recently it was demonstrated that free fermionic heterotic strings can produce models with solely the minimal supersymmetric standard model states in the low energy spectrum. This unprecedented result provides further strong evidence for the possibility that the true string vacuum shares some of the properties of the free fermionic models. Past free fermionic models have focused on several possible unbroken observable $\mathrm{SO}\left(10\right)\mathrm{}$ subgroups at the string scale, which include the flipped $\mathrm{SU}\left(5\right)\mathrm{}$ (FSU5), the Pati-Salam (PS) string models, and the string standard-like models (SLM). We extend this study to include the case in which the $\mathrm{SO}\left(10\right)\mathrm{}$ symmetry is broken to the left-right symmetric (LRS) gauge group, $\mathrm{SO}\left(10\right)\mathrm{}\to {\mathrm{SU}\left(3\right)\mathrm{}}_C\times {U\left(1\right)\mathrm{}}_{B-L}\times {\mathrm{SU}\left(2\right)\mathrm{}}_L\times {\mathrm{SU}\left(2\right)\mathrm{}}_R.$ We present several models of this type and discuss their phenomenological features. The most striking new outcome of the LRS string models, in contrast with the case of the FSU5, the PS, and the SLM string models, is that they can produce effective field theories that are free of Abelian anomalies. We discuss the distinction between the two types of free fermionic models which result in the presence, or absence, of an anomalous $U\left(1\right).$ As a counterexample we also present a LRS model that does contain an anomalous $U\left(1\right).$ Additionally, we discuss how in string models the standard model spectrum may arise from the three $16$ representations of $\mathrm{SO}\left(10\right),$ while the weak hypercharge does not have the canonical $\mathrm{SO}\left(10\right)\mathrm{}$ embedding.

We present a technique, the uniform asymptotic approximation, to construct accurate analytical solutions of the linear perturbations of inflation after quantum effects of the early Universe are taken into account, for which the dispersion relations generically become nonlinear. We construct explicitly the error bounds associated with the approximations and then study them in detail. With the understanding of the errors and the proper choice of the Liouville transformations of the differential equations of the perturbations, we show that the analytical solutions describe the exact evolution of the linear perturbations extremely well even when only the first-order approximation is used. As an application of the approximate analytical solutions, we calculate the power spectra and indices of scalar and tensor perturbations in the slow-roll inflation, and find that the amplitudes of the power spectra get modified due to the quantum effects, while the power spectrum indices remain the same as in the linear case.