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

Abstract

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 Yn≡|x−y|2nΔO&(O(x))n(O¯(y))n&in the limit where the operator insertion On has large total R-charge J= nΔO. We show that Yn has a nontrivial but universal asymptotic expansion at large J, of the form YnUnknowncharacter(0xF700)from"EuclidMathOne"(0x3D)J!(|NO|2π)2JJαY˜n,where Y˜ n approaches a constant as n → ∞, and NO is an n-independent constant describing on the normalization of the operator relative to the effective Abelian gauge coupling. The exponent α is a positive number proportional to the difference between the a-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 (Yn) , up to 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≃ 60 using the no-instanton approximation to the S4 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.