On the Large R-charge Expansion in mathcal{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 mathcal{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 {mathcal{Y}}_nequiv {left|x-yright|}^{2n{Delta}_{mathcal{O}}}leftlangle {left(mathcal{O}(x)right)}^n{left(overline{mathcal{O}}(y)right)}^nrightrangle in the limit where the operator insertion {mathcal{O}}^n has large total R-charge mathcal{J}=n{Delta}_{mathcal{O}} . We show that {mathcal{Y}}_n has a nontrivial but universal asymptotic expansion at large mathcal{J} , of the formYn=J!NO2π2JJαY˜n,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {\\mathcal{Y}}_n=\\mathcal{J}!{\\left(\\frac{\\left|{\\mathbf{N}}_{\\mathcal{O}}\\right|}{2\\uppi}\\right)}^{2\\mathcal{J}}{\\mathcal{J}}^{\\alpha }{\\tilde{\\mathcal{Y}}}_n, $$\\end{document}where {tilde{mathcal{Y}}}_n approaches a consstant as n → ∞, and {mathbf{N}}_{mathcal{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 {mathrm{mathcal{B}}}_n= log left({mathcal{Y}}_nright) , up to and including order log mathcal{J} against exact results from supersymmetric localization [2-5]. In the case of mathcal{N}=4 we find precise agreement and in the case mathcal{N}=2 we find reasonably good numerical agreement at mathcal{J}simeq 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 ssof the moduli space of vacua.