  We present a study of the impact of an uncertainty in the beam far side-lobe
knowledge on the measurement of the Cosmic Microwave Background $B$-mode signal
at large scale. It is expected to be one of the main source of systematic
effects in future CMB observations. Because it is crucial for all-sky survey
missions to take into account the interplays between beam systematic effects
and all the data analysis steps, the primary goal of this paper is to provide
the methodology to carry out the end-to-end study of their effect for a
space-borne CMB polarization experiment, up to the cosmological results in the
form of a bias $\delta r$ on the tensor-to-scalar ratio $r$. LiteBIRD is
dedicated to target the measurement of CMB primordial $B$ modes by reaching a
sensitivity of $\sigma \left( r \right) \leq 10^{-3}$ assuming $r=0$. As a
demonstration of our framework, we derive the relationship between the
knowledge of the beam far side-lobes and the tentatively allocated error budget
under given assumptions on design, simulation and component separation method.
We assume no mitigation of the far side-lobes effect at any stage of the
analysis pipeline. We show that $\delta r$ is mostly due to the integrated
fractional power difference between the estimated beams and the true beams in
the far side-lobes region, with little dependence on the actual shape of the
beams, for low enough $\delta r$. Under our set of assumptions, in particular
considering the specific foreground cleaning method we used, we find that the
integrated fractional power in the far side-lobes should be known at a level as
tight as $\sim 10^{-4}$, to achieve the required limit on the bias $\delta r <
1.9 \times 10^{-5}$. The framework and tools developed for this study can be
easily adapted to provide requirements under different design, data analysis
frameworks and for other future space-borne experiments beyond LiteBIRD.
