Relevance of operators in 4d N=1 theories

Question

I have a conceptual question about how to judge a given operator, \(\mathcal{O}\), in 4d \(\mathcal{N}=1\) theories is relevant or not.

In the literature, the criterion is given by \(R(\mathcal{O}) \leqslant 2\).  i.e., the operator is relevant or marginal if \(R(\mathcal{O}) \leqslant 2\). However, as I understand, an acceptable operator deformation for the theory must have \(R(\mathcal{O}) = 2\) so that \(R\)-symmetry is not broken.

How to understand why the \(R\)-charge, not the dimension of the operator, tells us that the operator is relevant, marginal or irrelevant? Are there any relations when the theory is in UV? How to resolve the "contradiction" that the \(R\)-charge for the superpotential should always be such that \(R=2\)?

Answer 1

For a general quantum field theory, it's true that the notions of relevant, marginal, irrelevant operators are defined in terms of dimensions of operators. In the special case of a supersymmetric theory, it is natural to study the special case of supersymmetric deformations and the familiar criterion in terms of dimensions can often be reformulated.

More precisely, in a unitary 4d N=1 superconformal field theory, there is a general ("BPS-like") bound, bounding below the dimension of an operator by a multiple of its R-charge. Operators inducing supersymmetric deformations are precisely the operators saturating this bound (chiral or antichiral). For such operators, dimension and R-charge determine each other and so the definition of relevant, marginal, irrelevant can be reformulated purely in tems of R-charge.

The condition R=2 is equivalent to the condition of marginality, itself equivalent to the preservation of R-symmetry. Indeed, R-symmetry is only guaranteed to be preserved in the superconformal case. In the non-superconformal case, R-symmetry is in general anomalous.