= Spectator multiplicities in gauge beta functions
{title2=$T_i(R_i)\prod_{j\ne i}\dim R_j$}
For a field in a tensor-product <representation> of a product <gauge group>, its contribution to the <beta function (physics)> of factor $i$ includes the dimensions of all other factors. Those indices label independent copies for that factor. For an Abelian factor, replace its <Dynkin index> by the squared charge. Thus a quark weak doublet contributes three copies to the weak beta function and two copies to the colour beta function; <hypercharge> sums include all six components. <Chirality> conjugation changes a <representation> and the sign of an Abelian charge, but not its quadratic index.
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