The results of a search for top squark (stop) pair production in final states with one isolated lepton, jets, and missing transverse momentum are reported. The analysis is performed with proton--proton collision data at $\sqrt{s} = 8$ TeV collected with the ATLAS detector at the LHC in 2012 corresponding to an integrated luminosity of $20$ fb$^{-1}$. The lightest supersymmetric particle (LSP) is taken to be the lightest neutralino which only interacts weakly and is assumed to be stable. The stop decay modes considered are those to a top quark and the LSP as well as to a bottom quark and the lightest chargino, where the chargino decays to the LSP by emitting a $W$ boson. A wide range of scenarios with different mass splittings between the stop, the lightest neutralino and the lightest chargino are considered, including cases where the $W$ bosons or the top quarks are off-shell. Decay modes involving the heavier charginos and neutralinos are addressed using a set of phenomenological models of supersymmetry. No significant excess over the Standard Model prediction is observed. A stop with a mass between $210$ and $640$ GeV decaying directly to a top quark and a massless LSP is excluded at $95$ % confidence level, and in models where the mass of the lightest chargino is twice that of the LSP, stops are excluded at $95$ % confidence level up to a mass of $500$ GeV for an LSP mass in the range of $100$ to $150$ GeV. Stringent exclusion limits are also derived for all other stop decay modes considered, and model-independent upper limits are set on the visible cross-section for processes beyond the Standard Model.
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