Fitting the Quark and Lepton Masses in String Theories

TL;DR

This paper investigates how certain orbifold string theories can reproduce observed quark and lepton masses and mixing angles, identifying specific orbifolds that fit experimental data and discussing limitations in fitting CKM parameters.

Contribution

It systematically explores orbifold constructions in string theory, identifying those that can accurately reproduce fermion mass patterns and constraining the shape of compactified space.

Findings

Feasible fit for $Z_3$, $Z_4$, $Z_6$--I, and possibly $Z_7$ orbifolds.

Compactified space size and shape are constrained by fermion mass data.

Other orbifolds are unlikely to fit the observed fermion masses.

Abstract

The capability of string theories to reproduce at low energy the observed pattern of quark and lepton masses and mixing angles is examined, focusing the attention on orbifold constructions, where the magnitude of Yukawa couplings depends on the values of the deformation parameters which describe the size and shape of the compactified space. A systematic exploration shows that for $Z_3$, $Z_4$, $Z_6$--I and possibly $Z_7$ orbifolds a correct fit of the physical fermion masses is feasible. In this way the experimental masses, which are low--energy quantities, select a particular size and shape of the compactified space, which turns out to be very reasonable (in particular the modulus $T$ defining the former is $T=O(1)$). The rest of the $Z_N$ orbifolds are rather hopeless and should be discarded on the assumption of a minimal $SU(3)\times SU(2)\times U(1)_Y$ scenario. On the other hand, due to stringy selection rules, there is no possibility of fitting the Kobayashi--Maskawa parameters at the renormalizable level, although it is remarked that this job might well be done by non--renormalizable couplings.