  We investigate the decoherence of the electron wavepacket in purely ballistic
one-dimensional systems described through the Luttinger liquid (LL). At a
finite temperature $T$ and long times $t$, we show that the electron Green's
function for a fixed wavevector close to one Fermi point decays as
$\exp(-t/\tau_F)$, as opposed to the power-law behavior occurring at short
times, and the emerging electron lifetime obeys $\tau_F^{-1}\propto T$ for
spinful as well as spinless electrons. For strong interactions, $(T\tau_F)\ll
1$, reflecting that the electron is not a good Landau quasiparticle in LLs. We
justify that fractionalization is the main source of electron decoherence for
spinful as well as spinless electrons clarifying the peculiar electron mass
renormalization close to the Fermi points. For spinless electrons and weak
interactions, our intuition can be enriched through a diagrammatic approach or
Fermi Golden rule and through a Johnson-Nyquist noise picture. We stress that
the electron lifetime (and the fractional quasiparticles) can be revealed from
Aharonov-Bohm experiments or momentum resolved tunneling. We aim to compare the
results with those of spin-incoherent and chiral LLs.
