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In general I would like to know as to known or what is/are the standard references about R-charge renormalization in supersymmetric theories. When does it do so and what is expected or known to be interesting.
A little more specifically I want to know whether something is particularly special if the R-charge of the scalar chiral superfield in some theory is seen to be flowing to values greater than 1/2 under the flow.
And if the R-charge of the scalar chiral superfield is flowing to values less than half then what does it physically imply? I have at times seen a vague argument along this line that if it is flowing to 0 asymptotically then that means that the theory is developing a continuum spectrum in that limit - which is supposed to be more surprising if the theory was defined on a compact space(-time?) to start with. But I don't understand the above argument any much more and would like to know of precise statements/derivations/references - and hopefully pedagogic ones from where a beginner in the field can learn!
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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