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In systems with competing interactions, continuous degeneracies can appearwhich are accidental, in that they are not related to any symmetry of theHamiltonian. Accordingly, the pseudo-Goldstone modes associated with thesedegeneracies are also unprotected. Indeed, through a process known as"order-by-quantum-disorder", quantum zero-point fluctuations can lift thedegeneracy and induce a gap for these modes. We show that this gap can beexactly computed at leading order in $1/S$ in spin-wave theory from the meancurvature of the classical and quantum zero-point energies - without the needto consider any spin-wave interactions. We confirm this equivalence throughdirect calculations of the spin-wave spectrum to $O(1/S^2)$ in a wide varietyof theoretically and experimentally relevant quantum spin models. We prove thisequivalence through the use of an exact sum rule that provides the requiredmixing of different orders of $1/S$. Finally, we discuss some implications forseveral leading order-by-quantum-disorder candidate materials, clarifying theexpected pseudo-Goldstone gap sizes in Er$_2$Ti$_2$O$_7$ andCa$_3$Fe$_2$Ge$_3$O$_{12}$.

Pseudo-Goldstone gaps and order-by-quantum-disorder in frustrated magnets