  We compute the effects of electronic interactions on gapless spin-3/2
excitations that in a noninteracting system emerge at a bi-quadratic touching
of Kramers degenerate valence and conduction bands, known as Luttinger
semimetal. This model can describe the low-energy physics of HgTe, gray-Sn, 227
pyrochlore iridates and half-Heuslers. For the sake of concreteness we only
consider the short-range components of the Coulomb interaction. By combining
mean-field analysis with a renormalization group (RG) calculation, we construct
multiple cuts of the global phase diagram of interacting spin-3/2 fermions at
zero and finite temperature and chemical doping. Such phase diagrams display a
rich confluence of competing orders, among which rotational symmetry breaking
nematic insulators and time reversal symmetry breaking magnetic orders are the
prominent excitonic phases. We also show that even repulsive interactions can
be conducive for both s-wave and d-wave pairings. The reconstructed band
structure inside the ordered phases allows us to organize them according to the
energy (entropy) gain in the following (reverse) order: s-wave pairing, nematic
phases, magnetic orders and d-wave pairings, at zero chemical doping. But, the
paired states are always energetically superior over the excitonic ones for
finite doping. The phase diagrams obtained from the RG analysis show that an
ordered phase with higher energy (entropy) gain is realized at low (high)
temperature. In addition, we establish a "selection rule" between the
interaction channels and the resulting ordered phases, suggesting that
repulsive interactions in the magnetic (nematic) channels are conducive for the
nucleation of d-wave (s-wave) pairing. The proposed methodology can shed light
on the global phase diagram of other strongly interacting multi-band systems,
such as doped Dirac semimetal, topological insulators and the like.
