I have a left-handed ##SU(2)## lepton doublet:(adsbygoogle = window.adsbygoogle || []).push({});

##

\ell_L = \begin{pmatrix} \psi_{\nu,L} \\ \psi_{e,L} \end{pmatrix}.

##

I want to know its transformation properties under conjugation and similar 'basic' transformations: ##\ell^{\dagger}_L, \bar{\ell}_L, \ell^c_L, \bar{\ell}^c_L## and the general left and right projections of ##\ell##. I've noticed there is ambiguity in what is meant with ##\ell_L##: whether it's the projection or the part with left-handed chirality (when taking the left projection of a Dirac spinor ##\bar\psi## one actually gets a spinor with right-handed helicity).

Naively, I thought that one just conjugates the components inside the doublet, for instance: ##\ell^c_L = \begin{pmatrix} \psi^c_{\nu,L} \\ \psi^c_{e,L} \end{pmatrix}##, but I've seen a book write ##\ell^c_L = - i \sigma^2 \begin{pmatrix} \psi^c_{\nu,L} \\ \psi^c_{e,L} \end{pmatrix}##, ##\sigma^2## is the second Pauli matrix, but I can't see where this comes from. Same for the rest of the conjugation transformations mentioned above, but no-one seems to write all of them down at once to see whether this is a conventional question or not.

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# SU(2) lepton doublet conjugation rules

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