I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is
$u^c d^c \tilde d^c$
where $\tilde d$ is the scalar superpartner of down quark. Now, being a scalar, this field doesn't transform under Lorentz transformation. This means that the term $u^c d^c$ is Lorentz invariant. Being $u$ and $d$ 4-component Dirac spinor this has to be read as
in order to proper contract rows and columns.
This means that also $u^T d$ should be Lorentz invariant...
However, Lorentz invariant are build with bar spinors, i.e.
$\bar \psi \psi$ is Lorentz invariant
while I don't see how
$\psi^T \psi$ can be Lorentz invariant.. Clearly I am missing something really basic here..