Lorentz invariance from Dirac spinor

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

$$(u^c)^T d^c$$

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.