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No. In high-energy physics it is typically the mass of a particle which is protected. Because of renormalization, the parameters of the Lagrangian change their values (“running coupling constants”), so the observed mass of a particle should “naturally” be on the order of the energy scale of the theory. Some masses are much smaller. The prime example is the photon mass. It is zero because of gauge symmetry (a mass term for the photon would break gauge symmetry). Another typical example is the pion, which has a small mass because of an approximate symmetry.