Saturating the Bound on the Scale of Fermion Mass Generation

Recently, Jäger andWillenbrock have shown that the Appelquist and Chanowitz bound on the scale of top-quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for f f̄ → VLVL and VLVL → VLVL scattering and derive the conditions on the parameters required for f f̄ → VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. ∗ e-mail addresses: [email protected] Appelquist and Chanowitz [1] derived an upper bound on the scale of fermion mass generation by examining the inelastic scattering amplitude f f̄ → VLVL, where VL denotes longitudinally polarized W or Z gauge bosons. In the absence of the Higgs boson, or another dynamics responsible for generating fermion mass, this amplitude grows with increasing center-of-mass energy. This tree-level amplitude would ultimately violate unitarity at a sufficiently high energy Λf . Therefore, one concludes that the scale associated with fermion mass generation is bounded by Λf . The strictest bound for a fermion with mass mf is obtained from the spin-zero, weak-isosinglet, color-singlet amplitude [2] Λf < 8πv √ 3Ncmf , (1.1) where v = 246 GeV and Nc is the number of colors (3 for quarks and 1 for leptons). The strongest bound occurs for the top quark. With mt ≈ 175 GeV, the upper bound (1.1) on the scale of top mass generation is Λt ≈ 3 TeV. On the other hand, in the absence of a Higgs boson or some other dynamics responsible for electroweak gauge-boson mass generation, the elastic scattering amplitude VLVL → VLVL grows quadratically with center-of-mass energy. Therefore, one concludes that the scale associated with gauge boson mass generation is bounded by a scale ΛEW where the elastic scattering amplitude would violate unitarity. The strictest bound is obtained from the spin-zero, weak-isosinglet scattering amplitude [3, 4, 2] ΛEW < √ 8πv ≈ 1.2TeV . (1.2) Since ΛEW is less than Λt, it is not clear that the bound (1.1) is relevant [1, 5]. The physics responsible for unitarizing the elastic gauge boson scattering amplitude may unitarize the inelastic tt̄ → VLVL amplitude, as happens in the standard one-doublet Higgs model. Recently, Jäger andWillenbrock [6] have shown that the Appelquist and Chanowitz [1] bound (1.1) on the scale of top quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for f f̄ → VLVL and VLVL → VLVL scattering and derive the conditions on the parameters required for f f̄ → VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. Consider the general potential in a two-Higgs model written in the form [7] V (φ1, φ2) = λ1(φ † 1φ1 − v 1/2)2 + λ2(φ † 2φ2 − v 2/2)2 + λ3 [ (φ1φ1 − v 1/2) + (φ † 2φ2 − v 2/2) ]2 + λ4 [ (φ1φ1)(φ † 2φ2)− (φ † 1φ2)(φ † 2φ1) ]