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In a old paper, Becca Asquith argues that it is possible to prove that if the SU(2)xU(1) sector of the standard model is chiral, then the SU(3)xU(1) sector is vectorial, ie, that at least one of them must be of purely vectorial character.

But the argument there was based on building the Standard Model from Connes-Lott models. At that time, I heard some folklore that this was really a general result, probably arguable on the basis of anomalies. But I can not pinpoint a concrete statement. So, is it a mathematical fact, that colour must be a Vector kind of force because SU(2) is chiral? Or just lore?

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