  We revisit the sensitivity to non-resonant, heavy Majorana neutrinos $N$ in
same-sign $W^\pm W^\pm$ scattering at the $\sqrt{s}=13$ TeV LHC and its
high-luminosity upgrade. As a benchmark scenario, we work in the context of the
Phenomenological Type I Seesaw model, relying on a simulation up to
next-to-leading order in QCD with parton shower matching. After extensively
studying the phenomenology of the $pp\to\mu^\pm\mu^\pm j j$ process at the
amplitude and differential levels, we design a simple collider analysis with
remarkable signal-background separation power. At 95\% confidence level we find
that the squared muon-heavy neutrino mixing element $\vert V_{\mu N} \vert^{2}$
can be probed down to about $0.06-0.3 ~ (0.03-0.1)$ for $m_N = 1-10~{\rm TeV}$
with $\mathcal{L}=300$ fb$^{-1}~(3$ ab$^{-1})$. For heavier masses of $m_N =
20~{\rm TeV}$, we report sensitivity for $\vert V_{\mu N} \vert^{2}\gtrsim
0.5~(0.3)$. The $W^\pm W^\pm$ scattering channel can greatly extend the mass
range covered by current LHC searches for heavy Majorana neutrinos and
particularly adds invaluable sensitivity above a few hundred GeV. We comment on
areas where the analysis can be improved as well as on the applicability to
other tests of neutrino mass models.

High Energy Physics - Phenomenology;High Energy Physics - Experiment