It's not clear what you mean by "the R-charge." If you have a U(1) R-symmetry, you can make linear combinations of its charges and those of a non-R U(1) symmetry and get a new R-symmetry. So "values greater than 1/2" are not, generically, special.

At a **superconformal** fixed point, there is a special U(1) R-symmetry that is part of the superconformal algebra. In this case, R-charges are related to operator dimensions and so are constrained by unitarity bounds. Maybe you have this in mind. The literature on $a$-maximization might be the sort of thing you're looking for.

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