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The tree level potential for a scalar multiplet of `Yukawa fields' $Y$ forone type of quarks admits the promising vacuum configuration $<Y> \propto {\rmdiag}(0,0,1)$ that breaks spontaneously $SU(3)_L\times SU(3)_R$ flavoursymmetry. We investigate whether the vanishing entries could be lifted tononvanishing values by slightly perturbing the potential, thus providing amechanism to generate the Yukawa hierarchies. For theories where at the lowestorder the only massless states are Nambu-Goldstone bosons we find, as a generalresult, that the structure of the tree-level vacuum is perturbatively stableagainst corrections from scalar loops or higher dimensional operators. Wediscuss the reasons for this stability, and give an explicit illustration inthe case of loop corrections by direct computation of the one-loop effectivepotential 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 simplemodel in which spontaneous breaking of the flavour symmetry can give rise tothe fermion mass hierarchies.

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