The half-composite two Higgs doublet model and the relaxion

We study a new con fining gauge theory with fermions in a vectorial representation under the SM gauge group that allows for Yukawa interactions with the Higgs. If the fermion masses are smaller than the con finement scale this realizes a type I two Higgs doublet model where a composite Higgs mixes with the elementary Higgs. This class of models interpolates between an elementary and a composite Higgs and has interesting phenomenology with potentially observable effects in collider physics, EDMs and SM couplings but very weak bounds from indirect searches. The very same framework can be used to realize the cosmological relaxation of the electro-weak scale recently discussed in the literature.