  Here we show how, in the ultra-strongly-coupled spin-boson model, apparently
unphysical "Matsubara modes" are required not only to regulate detailed
balance, but also to arrive at a correct and physical description of the
non-perturbative dynamics and steady-state. In particular, in the
zero-temperature limit, we show that neglecting the Matsubara modes results in
an erroneous emission of virtual photons from the collective ground state. To
explore this difficult-to-model regime we start by using a non-perturbative
hierarchical equations of motion (HEOM) approach, based on a partial fitting of
the bath correlation-function which takes into account the infinite sum of
Matsubara frequencies using only a biexponential function. We compare the HEOM
method to both a pseudo-mode model, and the reaction coordinate (RC) mapping,
which help explain the nature of the aberrations observed when Matsubara
frequencies are neglected. For the pseudo-mode method we present a general
proof of validity, which allows for negative Matsubara-contributions to the
decomposition of the bath correlation functions to be described by
zero-frequency Matsubara-modes with non-Hermitian coupling to the system. The
latter obey a non-Hermitian pseudo-Schr\"odinger equation, ultimately
justifying why superficially unphysical modes can give rise to physical system
behavior.
