A Lorentz invariance from Dirac spinor

d8586
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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..
 
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If you want people to help you, I think it's a good idea to share some references. Naively, I don't see how you term can be Lorentz-invariant either, but without reference or context that's all I can say.
 
You are right, sorry. I am looking for example at the lower right box of Tab. 3 of https://arxiv.org/pdf/1008.4884.pdf
In the B-violating operator I have terms that goes (neglecting colour and SU(2) indices, and noting that Dirac indices are always contracted within the brakets) like

$$d^T C u$$

and similar. This is what I meant
 

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