A tale of twin Higgs: natural twin two Higgs doublet models

In original twin higgs model, vacuum misalignment between electroweak and new physics scales is realized by adding explicit $$ {mathrm{mathbb{z}}}_2 $$ breaking term. introducing additional twin higgs could accommodate spontaneous $$ {mathrm{mathbb{z}}}_2 $$ breaking, which explains origin of this misalignment. we introduce a class of twin two higgs doublet models with most general scalar potential, and discuss general conditions which trigger electroweak and $$ {mathrm{mathbb{z}}}_2 $$ symmetry breaking. various scenarios on realising the vacuum misalignment are systematically discussed in a natural composite two higgs double model framework: explicit $$ {mathrm{mathbb{z}}}_2 $$ breaking, radiative $$ {mathrm{mathbb{z}}}_2 $$ breaking, tadpole-induced $$ {mathrm{mathbb{z}}}_2 $$ breaking, and quartic-induced $$ {mathrm{mathbb{z}}}_2 $$ breaking. we investigate the higgs mass spectra and higgs phenomenology in these scenarios.