New solution for dynamical symmetry breaking with top and bottom quark condensates

Abstract

Starting from a general $\rm SU_2\times U_1$ invariant interaction Lagrangian ${\cal L}_{\rm int}$ with four-fermion interactions between top $t$ and bottom $b$ quarks and working in the bubble approximation, we get a single Higgs as a bound state of quark pairs by allowing both $\langle \bar tt\rangle$ and $\langle\bar bb\rangle$ to be nonzero and the $t$ and $b$ states to mix. We find relations between the three four-fermion couplings $g_t, g_b,$ and $g_{tb}$ and show the new result that they may all be finite, with $gΛ^2\gg1$, where $Λ$ is the cutoff. Thus the dimensionless couplings $g' = gΛ^2$ correspond to strong interactions. Previous work with either one or more quark condensates found a fine-tuning condition giving $gΛ^2\sim$ O($1$). The Higgs mass $m_H$ is approximately the single-vev value $m_H\approx 2m_t$, and the quark mass ratio is $m_t/m_b\approx g_b/g_{tb}$. There is a new symmetry of ${\cal L}_{\rm int}$, corresponding to a flat direction in the space of composite states. Breaking this symmetry by turning on one small eigenvalue of the coupling matrix turns on the quark masses and introduces the massive Higgs state, so that $m_b,m_t,m_H\ll Λ$ is natural.