  In his 1977 paper on vacuum decay in field theory: The Fate of the False
Vacuum, Coleman considered the problem of a single scalar field and assumed
that the minimum action tunnelling field configuration, the bounce, is
invariant under O(4) rotations in Euclidean space. A proof of the O(4)
invariance of the bounce was provided later by Coleman, Glaser, and Martin
(CGM), who extended the proof to $N>2$ Euclidean dimensions but, again,
restricted non-trivially to a single scalar field. As far as we know a proof of
O($N$) invariance of the bounce for the tunnelling problem with multiple scalar
fields has not been reported, even though it was assumed in many works since,
being of phenomenological interest. We make progress towards closing this gap.
Following CGM we define the reduced problem of finding a field configuration
minimizing the kinetic energy at fixed potential energy. Given a solution of
the reduced problem, the minimum action bounce can always be obtained from it
by means of a scale transformation. We show that if a solution of the reduced
problem exists, then it and the minimum action bounce derived from it are
indeed O($N$) symmetric.
