Reparameterization of the scalar potential in the two Higgs doublet model

In the two Higgs doublet model with no additional symmetries in the scalar sector (different from the gauge and Lorentz symmetries), it is customary to reparameterize the model by rotating the scalar doublets such that one of the vacuum expectation values vanishes. It is well known that the Yukawa sector of the model is unaffected by such transformation. Notwithstanding, since the Higgs potential must also be transformed, it is necessary to show that such sector is also unaltered in its physical content. We demostrate that the physical content of the potential is invariant even when the charge conjugation symmetry is demanded.