Mass and Width of a Heavy Higgs Boson

The gauge dependence of the Higgs-boson mass and width in the on-shell scheme of renormalization is studied in the heavy-Higgs-boson approximation. The corresponding expansions in the pole scheme are analyzed adopting three frequently employed parametrizations. The convergence properties and other theoretical features of the on-shell and pole expansions, as well as their relative merits, are discussed. PACS numbers: 11.15.Bt, 12.15.Lk, 14.80.Bn ∗Permanent address: Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA. There exists a significant and interesting literature concerning the analysis of the mass and width of a heavy Higgs boson, both in the pole and on-shell schemes of renormalization [1,2,3,4,5]. The theoretical results may be conveniently expressed as expansions in λ/(2π) = GM H , where MH and λ are the mass and quartic coupling of the Higgs boson and G = GF/(2π √ 2). Calling g the SU(2) coupling, in the heavy-Higgs approximation (HHA), the limit g,MW ,MZ → 0 with G ∝ g/M W and λ held fixed is employed, and the top-quark and other fermionic contributions are neglected. In the HHA, the Higgsboson width and the relation between the on-shell and pole masses are known through O(λ), i.e., in the next-to-next-to-leading order (NNLO) [2,3,4,5]. Recently, however, it has been emphasized that, in the on-shell scheme, both the Higgs-boson mass and width are gauge-dependent quantities [6]. In this letter, we re-examine the on-shell-scheme expansions in the HHA, with particular emphasis on the issue of gauge dependence. We also re-analyze the pole-scheme expansions adopting three different, frequently employed parametrizations, and discuss their convergence properties, as well as other theoretical features. Calling M0 the bare mass and A(s) the self-energy, the on-shell mass M and width Γ of the Higgs boson are given by M = M 0 +ReA(M), MΓ = − ImA(M ) 1−ReA′(M2) . (1) Instead, in the pole scheme, one considers the complex-valued position of the propagator’s pole [7], s̄ = M 0 + A(s̄). (2) Given s̄, there is no unique way to define the pole mass and width. Two frequently employed parametrizations are s̄ =m 2 − im2Γ2, (3) s̄ = ( m3 − i 2 Γ3 2 , (4) with m2, Γ2 or m3, Γ3 identified with the mass and width of the unstable particle. A third definition is s̄ = m 1 − im1Γ1 1 + Γ21/m 2 1 (5) or, equivalently, m1 = √ m22 + Γ 2 2, Γ1 = m1 m2 Γ2. (6) In the Z-boson case, Eq. (6) leads, to very good approximation, to a Breit-Wigner resonance amplitude with an s-dependent width, and it has been shown that the m1 definition can be identified with the Z-boson mass measured at LEP [8]. An important property of s̄ and, therefore, also of mi and Γi (i = 1, 2, 3), is that they are gauge-invariant quantities.