  We study the four-top final state at the LHC as a probe for New Physics (NP)
effects due to new particles that couple predominantly to the top quark and
whose masses are below the top-quark-pair production threshold. We consider
simple NP models containing a new particle with either spin 0, spin 1, or spin
2, and find benchmark points compatible with current experimental results. We
find that interference effects between NP and QED amplitudes can be large,
pointing out the necessity of NLO contributions to be explicitly computed and
taken into account when NP is present. We examine kinematic differences between
these models and the Standard Model (SM) at the parton level and the
reconstructed level. In the latter case, we focus on events selected requiring
two same-sign leptons and multiple jets. We investigate how the different
Lorentz structure of the light NP affects the kinematic hardness, the
polarization, the spin correlations, and the angular distributions of the
parton-level and/or final-state particles. We find that spin-2 light NP would
be identified by harder kinematics than the SM. We also show that the angular
separation between the same-sign leptons is a sensitive observable for spin-0
NP. The spin-0 and spin-2 NP cases would also yield a signal in $t\bar t
\gamma\gamma$ with the invariant mass of the photons indicating the mass of the
new particle. The spin-1 NP would be identified through an excess in four-top
signal and slight or not modification in other observables, as for instance the
lack of signal in $t\bar t \gamma\gamma$ due to the Landau-Yang theorem. We
comment on the opportunities that would open from the kinematic reconstruction
of some of the top quarks in the four-top state. Our results provide new
handles to probe for light top-philic NP as part of the ongoing experimental
program of searches for four-top production at the LHC Run 2 and beyond.
