Theoretical Constraints on the Higgs Sector

• Nambu-Goldstone Bosons

• Equivalence Theorem

• Unitarity Constraints

• Constraints from the Renormalisation Group

• Indirect Constraint from Electroweak Precision Observables

• Theoretical considerations have provided and continue to provide powerful constraints on Higgs physics, both in the Standard Model and beyond.

• The Nambu-Goldstone theorem predicts the existence of massless scalar particles in the presence of spontaneously broken symmetries.

• The equivalence theorem connects the scattering amplitudes of highly-boosted massive gauge bosons with those of the Nambu-Goldstone bosons. • The unitarity constraint provides an upper bound on the Higgs mass $m_h$ (or equivalently the self coupling $\lambda$), which was known to be within the reach of the LHC before the discovery of the Higgs boson.

• The electroweak precision observables provide an indirect determination of the Higgs mass, which was a powerful constraint on the Higgs sector of the Standard Model even prior to the discovery of the Higgs boson at the LHC.

• The degree of fine-tuning in the Higgs sector is used to set (subjective) upper bounds on the scale $\Lambda$ at which beyond-the-Standard Model physics is expected to emerge.