Physics Asked on February 7, 2021
J.-P. Derendinger writes in his book the following lagrangian density for free fermions in chiral decomposition:
begin{equation}
isum_{n=1}^3 left[ overline{Psi}^{(n)}_{Q,j,alpha} gamma^mupartial_mu Psi^{(n),j,alpha}_Q +overline{Psi}_{U^c}^{(n),j} gamma^mupartial_mu Psi^{(n)}_{U^c,j} +overline{Psi}_{D^c}^{(n),j} gamma^mupartial_mu Psi^{(n)}_{D^c,j}+overline{Psi}^{(n)}_{L,alpha} gamma^mupartial_mu Psi^{(n),alpha}_L + overline{Psi}^{(n)}_{E^c} gamma^mupartial_mu Psi^{(n)}_{E^c} right],
end{equation}
where $n$ is the generation index, $j$ the color index, $alpha$ the weak isospin index and $c$ is for "charge conjugate". $Psi^{(n),j,alpha}_Q$ for example correspond to the components of the isospin doublet of left-handed quarks. I have strong doubts about his use of the charge conjugate (which he calls "anti-particle", and does not correspond to the second component of the bispinors). Indeed I think he should have wrote, for example, $Psi^{(n)}_{U,R,j}$ for right-handed Ups quarks instead of $Psi^{(n)}_{U^c,j}$, and so for the other charge conjugates. Am I misunderstanding something trivial about this lagrangian density or did J.-P. Derendinger wrote something wrong in his book?
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