On the Large R-charge Expansion in N=2 Superconformal Field Theories

In this note we study two point functions of Coulomb branch chiral ring elements with large R-charge, in quantum field theories with N = 2 superconformal symmetry in four spacetime dimensions. Focusing on the case of one-dimensional Coulomb branch, we use the effective-field-theoretic methods of [1], to estimate the two-point correlation function Y-n vertical bar x-y vertical bar(2n Delta O) <(O(x))(n) ((O) over bar (y))(n))> in the limit where the operator insertion O-n has large total R-charge J = n Delta(O). We show that Y-n has a nontrivial but universal asymptotic expansion at large J, of the form Y-n = J!(vertical bar N-O vertical bar/2 pi)(2J) J(alpha) (Y) over tilde (n), where (Y) over tilde (n) approaches a constant as n -> infinity, and N-O is an n-independent constant describing on the normalization of the operator relative to the effective Abelian gauge coupling. The exponent alpha is a positive number proportional to the difference between the alpha-anomaly coefficient of the underlying CFT and that of the effective theory of the Coulomb branch. For Lagrangian SCFT, we check our predictions for the logarithm B-n = log(Y-n), upto and including order log(J) against exact results from supersymmetric localization [2-5]. In the case of N = 4 we find precise agreement and in the case N = 2 we find reasonably good numerical agreement at J similar or equal to 60 using the no-instanton approximation to the S-4 partition function. We also give predictions for the growth of two-point functions in all rank-one SCFT in the classification of [6-9]. In this way, we show the large-R-charge expansion serves as a bridge from the world of unbroken superconformal symmetry, OPE data, and bootstraps, to the world of the low-energy dynamics of the moduli space of vacua.