Large-Nand vacuum alignment in top-color models

Top-color and top-color-assisted technicolor provide examples of dynamical electroweak symmetry breaking which include top-quark condensation, thereby naturally incorporating a heavy top quark. In this paper we discuss the roles of the Nambu–Jona-Lasinio (NJL) and large-$N$ approximations often used in phenomenological analyses of these models. We show that, in order to provide for top-quark condensation but not bottom-quark condensation, the top-color coupling must be adjusted to equal the critical value for chiral symmetry breaking up to $\mathcal{O}\mathrm{}(1/N)\mathrm{}$ in any theory in which the isospin-violating “tilting” interaction is a U(1) gauge interaction. A consequence of these considerations is that the potentially dangerous “bottom pions” are naturally light. We also show that the contributions to $\rho -1$ previously estimated are of leading order in $N,$ are not included in the usual NJL analysis, and are the result of “vacuum alignment.”