Lorentz-violating supersymmetric quantum field theories

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 nontrivial 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 Lorentz-violating extension of the standard model which is renormalizable by weighted power counting, the Lorentz fine tuning problem does not seem solvable by the requirement of supersymmetry.