  We consider a three-dimensional quantum antiferromagnet in the vicinity of a
quantum critical point separating the magnetically ordered and the magnetically
disordered phases. A specific example is TlCuCl$_3$ where the quantum phase
transition can be driven by hydrostatic pressure and/or by external magnetic
field. As expected two transverse and one longitudinal magnetic excitation have
been observed in the pressure driven magnetically ordered phase. According to
the experimental data, the longitudinal magnon has a substantial width, which
has not been understood and has remained a puzzle. In the present work, we
explain the mechanism for the width, calculate the width and relate value of
the width with parameters of the Bose condensate of magnons observed in the
same compound. The method of an effective quantum field theory is employed in
the work.
