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We investigate Planck scale boundary conditions on Higgs quartic interactionsand their $\beta$-functions in the SM augmented by an additional complexscalar. We use renormalisation group running at two-loops, and include boththeoretical and experimental constraints. We find that the boundary condition$\lambda=\beta_\lambda=0$ at the Planck scale is compatible with the currentHiggs and top mass measurements, but requires additional scalars lighter thanabout $600\,$GeV.

High scale boundary conditions with an additional complex singlet