Srednicki — Quantum Field Theory · Problem 96.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · mssm-higgs-sector

Problem

a) Write the mass terms for the Higgs scalars as LHiggs mass=−m12H†H−m22Hˉ†Hˉ−m32εij(HˉiHj+h.c.),(96.6)\mathcal{L}_{\text{Higgs mass}} = -m_1^2 H^\dagger H - m_2^2 \bar{H}^\dagger \bar{H} - m_3^2 \varepsilon^{ij}(\bar{H}_i H_j + \text{h.c.}) , \tag{96.6} and compute the quartic terms in HH and Hˉ\bar{H} that arise from eliminating the auxilliary SU(2) and U(1) DD fields. b) Find the conditions on the mass parameters in eq. (96.6) in order for the potential to be bounded below. c) Find the conditions on the mass parameters in eq. (96.6) in order to have spontaneous breaking of the SU(2) ×\times U(1) symmetry. d) Show that there are five physical Higgs particles, two charged and three neutral. e) Let tan⁡β≡vˉ/v\tan\beta \equiv \bar{v}/v be the ratio of the two Higgs VEVs. Show that tan⁡β+cot⁡β=(m12+m22)/m32.(96.7)\tan\beta + \cot\beta = (m_1^2 + m_2^2)/m_3^2 . \tag{96.7}

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