Large hierarchies from approximate R symmetries

TL;DR

This work investigates how large natural hierarchies can emerge from small, perturbatively generated constants in supersymmetric theories. By analyzing both exact and approximate $ ext{U}(1)_R$ symmetries, the authors show that $F$-term conditions enforce $\langle \mathscr{W} \rangle=0$ in the exact case, while approximate symmetries broken at high order $N$ yield $\langle \mathscr{W} \rangle \sim \langle \phi \rangle^{N}$, producing a hierarchically small constant. They provide a concrete string-theoretic realization in heterotic orbifold MiniLandscape models where high-order discrete $R$-symmetries give rise to accidental $ ext{U}(1)_R$ symmetries, FI-term cancellation with $|\,\phi\,|<1$, and a KKLT-like effective superpotential $\mathscr{W}_{\text{eff}}=c+Ae^{-aS}$ with $c=\langle \mathscr{W} \rangle=10^{-\mathcal{O}(10)}$, enabling dilaton stabilization and realistic gauge couplings. They also discuss implications for the MSSM $\mu$ term and demonstrate robustness under supergravity. The findings point to a natural mechanism for linking Planck-scale physics to low-energy scales via hierarchically small superpotential constants, with potential wide impact on moduli stabilization and SUSY phenomenology.

Abstract

We show that hierarchically small vacuum expectation values of the superpotential in supersymmetric theories can be a consequence of an approximate R symmetry. We briefly discuss the role of such small constants in moduli stabilization and understanding the huge hierarchy between the Planck and electroweak scales.