  Models with classical scale symmetry, which feature radiative symmetry
breaking, generically lead to a supercooled first-order phase transition in the
early Universe resulting in a strong gravitational-wave signal, potentially
observable by LISA. This is thanks to the absence of mass terms in the
potential and the resulting logarithmic structure of the zero-temperature
effective potential. It is known that imposing a symmetry at classical level
does not prohibit breaking it by quantum corrections. In the case of scale
symmetry, a mass term can in principle appear through renormalisation. This is
not the case in the commonly used $\overline{\mathrm{MS}}$ or Coleman-Weinberg
schemes. In this work, we renormalise a scale-invariant model in the on-shell
scheme to check whether parameterising it with the physical masses will
introduce mass terms to the potential. We find that indeed mass terms appear
for an arbitrary choice of the physical masses. However, we formulate an
on-shell condition for radiative symmetry breaking, sufficient and necessary
for the cancellation of mass terms in the renormalised potential, yielding a
logarithmic potential needed for supercooled phase transitions.
