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 ≡ |x − y| 2n∆ O (O(x)) n (Ō(y)) n in the limit where the operator insertion O n has large total R-charge J = n∆ O . We show that Y n has a nontrivial but universal asymptotic expansion at large J , of the formwhereỸ n approaches a constant as n → ∞, and N O 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(Y n ), up to and including order log(J ) against exact results from supersymmetric localization [2][3][4][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 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][7][8][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.

Abstract:We consider the operator spectrum of a three-dimensional N = 2 superconformal field theory with a moduli space of one complex dimension, such as the fixed point theory with three chiral superfields X, Y, Z and a superpotential W = XY Z. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large R-charge J. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly J + 1, a fact that cannot be seen directly from the XY Z Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension J + 2 − c −3 J −3 + O(J −4 ), where c −3 is an unknown positive coefficient. The coefficient c −3 is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of c −3 does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large J, this proves the existence of scalar semi-short states at all values of J. Thus the combination of N = 2 superconformal symmetry with the large-J expansion is more powerful than the sum of its parts.