Effects of a real singlet scalar on Veltman condition

We revisit the fine-tuning problem in the Standard Model (SM) and show the modification in the Veltman condition by virtue of a minimally-extended particle spectrum with one real SM gauge singlet scalar field. We demand the new scalar to interact with the SM fields through Higgs portal only, and the new singlet to acquire a vacuum expectation value, resulting in a mixing with the CP-even neutral component of the Higgs doublet in the SM. The experimental bounds on the mixing angle are determined by the observed best-fit signal strength σ/σSMσ/σSM. While, the one-loop radiative corrections to the Higgs mass squared, computed with an ultraviolet cut-off scale Λ, come with a negative coefficient, the quantum corrections to the singlet mass squared acquire both positive and negative values depending on the parameter space chosen, which if positive might be eliminated by introducing singlet or doublet vector-like fermions. However, based upon the fact that there is mixing between the scalars, when transformed into the physical states, the tree-level coupling of the Higgs field to the vector-like fermions worsens the Higgs mass hierarchy problem. Therefore, the common attempt to introduce vector-like fermions to cancel the divergences in the new scalar mass might not be a solution, if there is mixing between the scalars.