# String Landscape and Fermion Masses

## TL;DR

This paper proposes that in string theory, probability distributions of physical quantities tend to peak at zero, explaining the smallness of parameters like the cosmological constant and fermion masses, and making testable predictions about neutrino masses.

## Contribution

It introduces a probabilistic framework where distributions of physical parameters in string theory naturally peak at zero, linking string landscape to observable fermion and neutrino masses.

## Key findings

- Distribution P(Λ) peaks sharply at Λ=0, explaining small cosmological constant.
- Fermion mass distributions diverge at zero, consistent with observed small masses.
- Predicts sum of neutrino masses to be approximately 0.0592 eV, with an upper bound of 0.066 eV.

## Abstract

Besides the string scale, string theory has no parameter except some quantized flux values; and the string theory Landscape is generated by scanning over discrete values of all the flux parameters present. We propose that a typical (normalized) probability distribution $P({\cal Q})$ of a physical quantity $\cal Q$ (with nonnegative dimension) tends to peak (diverge) at ${\cal Q}=0$ as a signature of string theory. In the Racetrack K\"ahler uplift model, where $P(\Lambda)$ of the cosmological constant $\Lambda$ peaks sharply at $\Lambda=0$, the electroweak scale (not the electroweak model) naturally emerges when the median $\Lambda$ is matched to the observed value. We check the robustness of this scenario. In a bottom-up approach, we find that the observed quark and charged lepton masses are consistent with the same probabilistic philosophy, with distribution $P(m)$ that diverges at $m=0$, with the same (or almost the same) degree of divergence. This suggests that the Standard Model has an underlying string theory description, and yields relations among the fermion masses, albeit in a probabilistic approach (very different from the usual sense). Along this line of reasoning, the normal hierarchy of neutrino masses is clearly preferred over the inverted hierarchy, and the sum of the neutrino masses is predicted to be $\sum m_{\nu} \simeq 0.0592$ eV, with an upper bound $\sum m_{\nu} <0.066$ eV. This illustrates a novel way string theory can be applied to particle physics phenomenology.