Regularization at the next-to-leading order in the top-mode standard model without gauge bosons

We study Higgs condensation H∼ttin the top-mode standard model at the next-to-leading (NTL) order in 1/Ncby calculating the effective potential as a function of a hard mass termσ0of the top quark. We include the effects of the third generation quarks, the Higgs and the Nambu–Goldstone fields, and the leading quantum chromodynamic effects, but not the effects of the transverse components of the electroweak gauge bosons. The resulting effective theory contains finite energy cutoff parameters (Λf, Λb) for the fermionic and the bosonic degrees of freedom. Condensation is supposed to take place at energiesΛ∼Λf∼Λb. The paper describes how to regularize the integrals over the fermionic momenta in a way free of momentum branching ambiguities and how to treat the terms of 1/Ncexpansion in a mutually consistent manner. This is achieved by the proper time approach, employing specifically the proper time cutoff (PTC) or a Pauli–Villars (PV) regulator. For comparison, we use also the covariant spherical (S) cutoff. On the other hand, for the bosonic momenta we have to use the covariant spherical cutoff. We discuss how to ensure the validity of the Goldstone theorem. Demanding that the NTL contributions not “wash out” the leading ones, we obtain rather low bounds for the cutoffs:Λ=O (1 TeV). The results for the corresponding cases with PTC, PV, and S regularization differ only marginally. Thus, in the described framework, the 1/Ncexpansion approach has a predictive power only ifΛ=O (1 TeV), a result largely independent of the regularization procedure.