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Before concluding that conductivity scales as $1 / T^2$, there's one element left to show. A non-infinite conductivity requires both a decay time and an energy loss mechanism. In the current form, the energy the quasiparticle looses goes back to the Fermi sea, effectively accelerating it. However, in order   Problem: energy conservation of qp-qp scattering. Solution: umklapp processes, whereby energy is transferred to the lattice, therefore you have dissipation via qp-qp scattering alone. \cite{pal2012}  \subsection{Quasiparticle--impurity scattering}