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The CKM, PMNS matrices are mathematically absolutely analogous, except that the values and even hierarchies of all the parameters are entirely different in the two cases. (Also, we don't know whether right-handed neutrinos exist and whether the effective Majorana masses may be derived from Dirac masses or something else.) But both matrices may be reduced to ...

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Following the convention of Peskin & Schroeder- transformation rule of Dirac spinors $\psi(x,t)$ under $\cal{C}$ and $\cal{P}$ are given by, \begin{eqnarray} \cal{C}\psi(t,x)\cal{C}^{\dagger} & = & -i(\bar{\psi}\gamma^{0}\gamma^{2})^{\textbf{T}}\\ \cal{C}\bar{\psi}(t,x)\cal{C}^{\dagger} & = & -i(\gamma^{0}\gamma^{2}\psi)^{\textbf{T}}\\ ...

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I am answering a question after some clarifications. If the gauge field maps the electron to a positron, it really connects the left-handed 2-spinor and right-handed 2-spinor in the Dirac's electron field into one multiplet. But that field is also a part of the $SU(2)_W$ doublet with the neutrinos. So the theory you are proposing wants to extend the ...

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Short answer: to accurately model reality. Long answer: The weak interaction has several peculiar properties: The $W$ bosons are vector bosons (so the weak theory is likely a gauge theory) The $W$ bosons have electric charge The $W$ bosons have mass. (The $Z$ boson hadn't been observed experimentally; it was a prediction of the SM) The $W$ bosons couple ...

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Detailed answer The thing is that you cannot really force in an a priori interpretation of the gauge group with which you extend your existing theory. You can only decide on the symmetries of your theory. So keeping things very general you start by trying to gauge the symmetry group $SU(2)$. This has three generators and so gives rise to three independent ...

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That cannot be true, because $$\left(\frac{1-\gamma_5}2\right)\left(\frac{1+\gamma_5}2\right)=0$$ EDIT: The youtube video from the comments is actually wrong. The correct expression is $$\bar\psi_L\gamma^\mu\psi_{R,\mu}=\bar\psi P_R\gamma^\mu P_R\psi_{,\mu}$$ From this, you should be able to show that this term vanishes.

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