  We study the possibility of constructing Lorentz-violating supersymmetric
quantum field theories under the assumption that these theories have to be
described by lagrangians which are renormalizable by weighted power counting.
Our investigation starts from the observation that at high energies
Lorentz-violation and the usual supersymmetry algebra are algebraically
compatible. Demanding linearity of the supercharges we see that the requirement
of renormalizability drastically restricts the set of possible
Lorentz-violating supersymmetric theories. In particular, in the case of
supersymmetric gauge theories the weighted power counting has to coincide with
the usual one and the only Lorentz-violating operators are introduced by some
weighted constant c that explicitly appears in the supersymmetry algebra. This
parameter does not renormalize and has to be very close to the speed of light
at low energies in order to satisfy the strict experimental bounds on Lorentz
violation. The only possible models with non trivial Lorentz-violating
operators involve neutral chiral superfields and do not have a gauge invariant
extension. We conclude that, under the assumption that high-energy physics can
be described by a renormalizable Lorentz-violating extensions of the Standard
Model, the Lorentz fine tuning problem does not seem solvable by the
requirement of supersymmetry.
