I would like to ask, to make it clearer to me, what does chirality means? and how can someone see whether his theory can be chiral or not? I think chirality in the standard model appears due to the fact that the left and right handed components are transformed not identically under the Lorentz Group (or in other words the Lorentz Group is isomorphic to the [itex]SU(2)_{L} \times SU(2)_{R}[/itex] One example I came across this question was when I first saw the compactification of a 5D Dirac Action to a 4D one + circle . In that case you end up with a non-chiral theory, and that was because (as it was illustrated to me) the mass term for the zero-th mode, would appear in the form: [itex] (\bar{\psi}_{R} \psi_{L} - \bar{\psi}_{L} \psi_{R}) [/itex] and due to the minus it was supposed to be non-chiral. The problem is solved once the compactification is done on an orbifold. However, there are also other cases in which I came across that the theory cannot be chiral. For example the extended supersymmetric theories ([itex]N>1[/itex]). In this case I am still unable to see how that can be true. Any clues? Also if you find a mistake in anything I wrote, feel free to correct me.
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