Yukawa hierarchies from spontaneous breaking of the $\bma{SU(3)_L\times SU(3)_R}$ flavour symmetry?

TL;DR

This paper explores how spontaneous breaking of a flavor symmetry can generate Yukawa hierarchies, analyzing the stability of vacuum configurations and proposing a model for fermion mass hierarchies.

Contribution

It demonstrates that the vacuum structure is perturbatively stable and introduces a simple model to produce Yukawa hierarchies through spontaneous symmetry breaking.

Findings

Vacuum configuration with zero entries is stable against loop corrections.

Hierarchical Yukawa structures can be achieved by extending the scalar sector.

Explicit calculation of the one-loop effective potential supports the stability analysis.

Abstract

The tree level potential for a scalar multiplet of `Yukawa fields' $Y$ for one type of quarks admits the promising vacuum configuration $<Y> \propto {\rm diag}(0,0,1)$ that breaks spontaneously $SU(3)_L\times SU(3)_R$ flavour symmetry. We investigate whether the vanishing entries could be lifted to nonvanishing values by slightly perturbing the potential, thus providing a mechanism to generate the Yukawa hierarchies. For theories where at the lowest order the only massless states are Nambu-Goldstone bosons we find, as a general result, that the structure of the tree-level vacuum is perturbatively stable against corrections from scalar loops or higher dimensional operators. We discuss the reasons for this stability, and give an explicit illustration in the case of loop corrections by direct computation of the one-loop effective potential of Yukawa fields. Nevertheless, a hierarchical configuration $<Y>\propto {\rm diag}(\epsilon',\epsilon,1)$ (with $\epsilon', \epsilon\ll 1$) can be generated by enlarging the scalar Yukawa sector. We present a simple model in which spontaneous breaking of the flavour symmetry can give rise to the fermion mass hierarchies.