Large-Nand vacuum alignment in top-color models

Abstract

Topcolor and topcolor-assisted technicolor provide examples of dynamical electroweak symmetry breaking which include top-condensation, thereby naturally incorporating a heavy top quark. In this note 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-condensation but not bottom-condensation, the top-color coupling must be adjusted to equal the critical value for chiral symmetry breaking up to O(1/N ) 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 ρ − 1 previously estimated are of leading-order in N , are not included in the usual NJL analysis, and are the result of "vacuum-alignment".

Higgs mass from compositeness at a multi-TeV scale

“…If these arise from a spontaneously broken gauge symmetry, then they are accompanied by LL and RR 4-fermion operators that contribute to the coefficient of the [46]. The ensuing correction to the T parameter, coming from diagrams with two or more fermion bubbles, is expected to be somewhat smaller than our result in eq.…”

Effective field theory of vacuum tilting

“…Thus this contribution does not factorize, and we cannot relate it to (11). Note also that the U(1) gauge coupling goes like 1/ √ N for large N [24] so the contribution of ( 14) to the Φ mass goes to zero as N → ∞. Note that the diagrams in Figs.…”

“…U(1) gauge boson exchange -beyond leading order in N Now finally we are ready to discuss the effect of U(1) gauge boson exchange beyond the leading order in N. This is important, because it is just such a nonleading effect that produces the "tilting" of the vacuum alignment that is crucial to the phenomenology of topcolor models [1,2,24]. We begin by discussing the situation for m > M where we first integrate out the U(1) gauge boson to obtain the 4-fermion operator (8).…”

Minimal composite Higgs model with light bosons

“…(2.13). The four-quark operators with left-left or right-right current-current structure do not contribute to the effective potential in the large-N c limit (though they do contribute to observables, most importantly to the ρ parameter [12,13], but these contributions are sufficiently small for M c above a few TeV [14]). All other four-quark operators which are invariant under the Standard Model gauge group and involve only the ψ L , t R and χ fields violate the global U(1) t × U(1) χ symmetry, and are not expected to be induced by heavy gauge boson exchanges.…”

“…Such a hierarchy arises if there is a second order phase transition in which a continuous variation of the gauge coupling induces a continuous variation of the scalar masses. There are various arguments, based in general on the large-N c limit, indicating that a spontaneously broken gauge group leads indeed to a second order phase transition [13,15]. In practice, since we do not require an exponential hierarchy, it is sufficient to have a weakly first order phase transition.…”