  In this work the existence of Borromean states has been discussed for bosonic
and fermionic cases in both the relativistic and non-relativistic limits from
the 3-momentum shell renormalization. With the linear bosonic model we checked
the existence of Efimov-like states in the bosonic system. In both limits a
geometric series of singularities are found in the 3-boson interaction vertex,
while the energy ratio is reduced by around 70\% in the relativistic limit
because of the anti-particle contribution. Motivated by the quark-diquark model
in heavy baryon studies, we have carefully examined the p-wave quark-diquark
interaction and found an isolated Borromean pole at finite energy scale. This
may indicate a special baryonic state of light quarks in high energy quark
matters. In other cases trivial results are obtained as expected. In
relativistic limit, for both bosonic and fermionic cases, potential Borromean
states are independent of the mass, which means the results would be valid even
in zero-mass limit as well.
