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The ferromagnetically polarized gapped XXZ spin chain is studied at lowtemperatures. Utilizing only the one- and two-magnon spectrums and focusing onthe magnon-creation contribution to the transverse dynamical susceptibility, werepresent the latter in the form of the Dyson equation. Then, following thewell known correspondence between the imaginary part of magnetic susceptibilityand dynamical structure factor, we get the low-temperature formula for themagnon-peak lineshape. The suggested approach is effective only if theprocesses related to magnon creations and to transitions from magnons tocoupled magnon pairs are energetically separated. As it is shown in the paper,such separation is inherent in the easy-axis chains with rather stronganisotropy. We present several plots and discuss their lineshapes. As thesupplemental result we obtain integral representations for thetemperature-dependent magnon resonance shift and the parameter which is usuallyassociated with the decay rate. The low-temperature behavior of the resonanceshift is studied in details. All calculations are performed up to {\itcontrollable} error $o({\rm e}^{-\beta E_{gap}})$.

Magnon peak lineshape in the transverse dynamical structure factor of a magnetically polarized easy-axis $XXZ$ chain at low temperatures