Higgs bosons in particle physics and in condensed matter

Higgs bosons - the amplitude modes - have been experimentally investigated in condensed matter for many years. An example is superfluid ³He-B, where the broken symmetry leads to 4 Goldstone modes and at least 14 Higgs modes, which are characterized by angular momentum quantum number J and parity (Zeeman splitting of Higgs modes with J=2⁺ and J=2^- in magnetic field has been observed in 80's). Based on the relation E _J+²+E_J-²=4Δ² for the energy spectrum of these modes, Yoichiro Nambu proposed the general sum rule, which relates masses of Higgs bosons and masses of fermions. If this rule is applicable to Standard Model, one may expect that the observed Higgs boson with mass M_H1=125 GeV has a Nambu partner - the second Higgs boson with mass M_H2=325 GeV. Together they satisfy the Nambu relation M_H1² + M_H2² = 4 M_top ², where M _top is the top quark mass. Also the properties of the Higgs modes in superfluid ³He-A, where the symmetry breaking is similar to that of the Standard Model, suggest the possible existence of two electrically charged Higgs particles with masses M _H+=M _H-∼245 GeV, which together obey the Nambu rule M _H+² + M_H-² = 4 M_top ². A certain excess of events at 325 GeV and at 245 GeV has been reported in 2011, though not confirmed in 2012 experiments. Besides, we consider the particular relativistic model of top - quark condensation that suggests the possibility that two twice degenerated Higgs bosons contribute to the Nambu sum rule. This gives the mass around 210 GeV for the Nambu partner of the 125 GeV Higgs boson. We also discuss the other possible lessons from the condensed matter to Standard Model, such as hidden symmetry, where light Higgs emerges as quasi Nambu-Goldstone mode, and the role of broken time reversal symmetry.