Fixed Points: Fermion Mass Predictions

Abstract

Familiar theoretical relationships exist amongst the masses of the light fermions in SU(5):¹

$$ \frac{{{m_d}}}{{{m_e}}} \approx \frac{{{m_s}}}{{{m_{\mu }}}} \simeq 4\pm \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} \quad \frac{{{m_b}}}{{{m_{\tau }}}} \simeq 2.75\pm .25 $$

These result from essentially three ingredients: (a) the Higgs mechanism which relates the fermion mass to the Higgs-Yukawa (HY) coupling constant by ${m_f} = {g_f} < {\phi^o} > /\sqrt {2} \approx {g_f} \cdot (175\quad GeV)$ (b) the renormalization group (RG) equations which describe the evolution of the (HY) couplings down from M_X to low energies and (c) the boundary conditions from SU(5), assuming a 5 and 24 of Higgs, which relate g_{−1/3 quark} (M_X) = g_{−1 lepton} (M_X) within a given generation. Thus one obtains:

$$ \frac{{^{m} - 1/3}}{{^{m} - 1}} = \frac{{^{g} - 1/{{3}^{{(\mu )}}}}}{{^{g} - {{1}^{{(\mu )}}}}} \approx {{\left( {\frac{{{{\alpha }_{3}}\left( \mu \right)}}{{{{\alpha }_{3}}\left( {{{M}_{X}}} \right)}}} \right)}^{{8/2{{b}_{0}}}}} + {{\left. {\left( {electroweak} \right)} \right|}_{{\mu \simeq {{m}_{{quark}}}}}} $$

(2)

and the numerical results of eq. (1) follow to one loop accuracy with Λ_s ~. 4 GeV. We note that ingredients (b) and (c) do not apply to models with composite Higgs bosons (e.g. ETC) or composite fermions on a scale of M′ < M_X since the RG equations demand pointlike particles over the entire range of the desert. Though the results for m_d/m_e and m_s/m_μ are questionable we note that they are qualitatively correct. We take the “prediction” for m_b/m_τ to be a successful result of SU(5) and we seek to extend the above analysis to much heavier objects; either a very heavy top-quark or a heavy fourth generation of quarks and leptons.