  The complete set of hallmarks of the three-dimensional antiferromagnet near
the quantum critical point has been recently observed in the spin dimer
compound TlCuCl$_3$. Nonetheless, the mechanism, responsible for several
distinct features of the experimental data, has remained a puzzle, namely: (i)
the paramagnons exhibit remarkable robustness to thermal damping and are stable
up to high temperatures, where $k_B T$ is comparable with the excitation
energy, (ii) the width to mass ratios of the high-temperature paramagnons are,
within the error bars, equal to that of the low-temperature amplitude (or
Higgs) mode. We propose such a mechanism and identify two principal factors,
contributing to the scaling between width to mass ratios of the paramagnon and
the amplitude mode: (i) the emergence of the thermal mass scale reorganizing
the paramagnon decay processes, and (ii) substantial renormalization of the
multi-magnon interactions by thermal fluctuations. The study is carried out for
the general case of a $D= 3 + 1$ quantum antiferromagnet within the framework
of the $\varphi^4$ model using the hybrid Callan-Symanzik + Wilson thermal
renormalization group method. Our approach is tested by demonstrating a good
quantitative agreement with available experimental data across the phase
diagram of TlCuCl$_3$.
