Fermion boson mass inequalities and composite models for quarks and leptons

We consider models where quarks and leptons are composites of fermions (ψ) and bosons (φ) bound via some "meta-color" gauge interactions with respect to which ψ and φ are (say) in the fundamental representation. Let F; S, and π be the lowest flavor non singlet bound states in the (j = 1 2 ψφ⁺; (0⁺)φ⁺φ and ψγ₅ψ(0^-) meta-color singlet channels. In some composite models the following inequality 2m_F > m_s + m_π follows from Schwartz type inequalities between fermionic and bosonic Green functions utilizing the approach introduced by Weingarten Vafa-Witten and Witten. Since there are no scalar and/or pseudo-scalars counterparts of the light fermions the class of viable composite models may be most severely constrained. We also indicate how the variational approach may yield many useful inequalities of the type m(xy) ≥ 1 2[m(xx) + m(yy)].