String Unification and Threshold Corrections

The interpretation of the apparent unification of gauge couplings within supersymmetric theories depends on uncertainties induced through heavy particle thresholds. While in standard grand unified theories these effects can be estimated easily, the corresponding calculations are quite complicated in string unified theories and do exist only in models with unbroken E6. We present results for heavy particle thresholds in more realistic models with gauge group SU(3) × SU(2) × U(1). Effects of Wilson line background fields as well as the universal part of the (rather mild) threshold corrections indicate a strong model dependence. We discuss the consequences of our results for the idea of string unification without a grand unified gauge group. ∗ Partially supported by the Deutsche Forschungsgemeinschaft and the EC under contract no. SC1–CT92–0789. An appealing concept for the extension of the successful standard model (SM) to higher energy scales is the idea of grand unified theories (GUTS). They provide a useful tool to gain knowledge about new physics up to the unification scale MX . The evolution of the gauge couplings according to the renormalization group equations (RGE) is determined by the particle spectrum below MX ; therefore the gauge couplings are sensitive detectors for the presence of each particle with nontrivial gauge quantum numbers. In the context of the minimal supersymmetric extension of the standard model (MSSM) the extrapolation of the low-energy data for the SU(3) × SU(2) × U(1) gauge couplings is consistent with grand unification at a scale of MX ≈ 2 · 10 16 GeV [1]. This agreement could serve as an argument for the existence of a unified gauge group as e.g. SU(5), SO(10) or E6 above the unification scale MX . Unfortunately the reliability of this result is weakened by the presence of uncertainties due to threshold effects at large scales [2]. In the presence of large group representations, e.g. a 75 in SU(5) models [3], these corrections can acquire a considerable size and disturb seriously this successful picture. Moreover there are deep conceptual problems peculiar to supersymmetric GUTS: the avoidance of too fast proton decay raises the well–known problem of doublet–triplet splitting leading to a revival of the mass hierarchy problem whose removal was one of the primary motivations for the introduction of supersymmetry. In addition the boundary conditions for the Yukawa couplings at MX forced by the group structure put serious constraints on the fermion mass spectrum. Finally these theories are to be embedded in a more complete theory at the natural high energy scale MP lanck where gravitational quantum effects can no longer be neglected. Amazingly superstring theories, the only known theories which consistently unify all known interactions including gravity, show likewise the feature of gauge coupling unification. At the string scale Mstring ≈ 0.7 · gstring · 10 18 GeV [4, 5] the gauge coupling constants of the various group factors are related to the gravitational coupling constant GN at tree level as follows [6]: g aka = 4πα GN = gstring , ∀ a , (1) where α is the string tension and ka is the Kac–Moody level of the group factor labeled by a. Below Mstring effective theories are an appropriate description of the low–energy physics and the couplings begin to evolve according to the RGE as in ordinary field theories. At the one–loop level the above equation is modified to [7, 4] 16π g2 a(μ) = ka 16π g string + ba ln ( M string μ2 )