  We investigate the linear response of an O(N) scalar quantum field theory
subject to external perturbations using the symmetry improved two particle
irreducible effective action formalism [A. Pilaftsis and D. Teresi, Nucl. Phys.
B874, 594 (2013)]. Despite satisfactory equilibrium behavior, we find a number
of unphysical effects at the linear response level. Goldstone boson field
fluctuations are over-determined, with the only consistent solution being to
set the fluctuations and their driving sources to zero, except for momentum
modes where the Higgs and Goldstone self-energies obey a particular
relationship. Also Higgs field fluctuations propagate masslessly, despite the
Higgs propagator having the correct mass. These pathologies are independent of
any truncation of the effective action and still exist even if we relax the
over-determining Ward identities, so long as the constraint is formulated
O(N)-covariantly. We discuss possible reasons for the apparent incompatibility
of the constraints and linear response approximation and possible ways forward.
