Supersymmetry Breaking and Determination of the Unification Gauge Coupling Constant in String Theories

TL;DR

This work analyzes how gaugino condensation in the hidden sector of string-derived supergravity can simultaneously generate hierarchical supersymmetry breaking and stabilize the dilaton $S$, whose real part fixes the unification gauge coupling via ${\rm Re}\,S\sim g_{gut}^{-2}$. Using a modular-invariant framework with a holomorphic superpotential $W$ and gauge kinetic function $f$, the authors show that pure Yang–Mills condensation fails to produce realistic $S_R$, while the inclusion of hidden matter enables a broad class of two-condensate scenarios in which $S_R\approx 2$ and the gravitino mass $m_{3/2}$ falls in the phenomenologically viable range around $10^3$ GeV, without fine-tuning. A remarkable finding is that good values of $S_R$ and $m_{3/2}$ are correlated and largely robust against variations in the Green–Schwarz parameter $\delta^{GS}$, with the modulus $T$ stabilizing near $T\approx 1.23$ and duality broken spontaneously. The analysis covers untwisted and twisted matter, accounts for $\delta^{GS}$ corrections, and provides an exhaustive classification of viable hidden-sector configurations, highlighting the central role of multiple condensates in achieving realistic dilaton stabilization in string theories.

Abstract

We study in a systematic and modular invariant way gaugino condensation in the hidden sector as a potential source of hierarchical supersymmetry breaking and a non--trivial potential for the dilaton $S$ whose real part corresponds to the tree level gauge coupling constant (${\rm Re}\ S\sim g_{gut}^{-2}$). For the case of pure Yang--Mills condensation, we show that no realistic results (in particular no reasonable values for ${\rm Re}\ S$) can emerge, even if the hidden gauge group is not simple. However, in the presence of hidden matter (i.e. the most frequent case) there arises a very interesting class of scenarios with two or more hidden condensing groups for which the dilaton dynamically acquires a reasonable value (${\rm Re}\ S\sim 2$) and supersymmetry is broken at the correct scale ($m_{3/2}\sim 10^3\ GeV$) with no need of fine--tuning. Actually, good values for ${\rm Re}\ S$ and $m_{3/2}$ are correlated. We make an exhaustive classification of the working possibilities. Remarkably, the results are basically independent from the value of $δ^{GS}$ (the contributions from the Green--Schwarz mechanism). The radius of the compactified space also acquires an expectation value, breaking duality spontaneously.